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May 28, 2023, 11:44:28 AM5/28/23

to

we have

∀n ∈ ℕ: 1/n - 1/(n+1) = 1/(n(n+1)) > 0 .

note the universal quantifier. This holds for every unit fraction. Thus for every unit fraction (unrevealed or invisible or ...) there is another unit fraction closer to zero (the gap is finite.),

Thus the is no first unit fraction.

∀n ∈ ℕ: 1/n - 1/(n+1) = 1/(n(n+1)) > 0 .

note the universal quantifier. This holds for every unit fraction. Thus for every unit fraction (unrevealed or invisible or ...) there is another unit fraction closer to zero (the gap is finite.),

Thus the is no first unit fraction.

May 28, 2023, 4:05:32 PM5/28/23

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Message has been deleted

May 28, 2023, 4:22:37 PM5/28/23

to

Not 'to its limit'.

Only TOWARDS its limit.

It never arrives 'at zero'.'

Message has been deleted

May 28, 2023, 4:38:05 PM5/28/23

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basis, right?

i[0] = 1/1

i[1] = 1/2

i[2] = 1/3

i[3] = 1/4

i[4] = 1/5

....

May 28, 2023, 4:39:46 PM5/28/23

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Fair enough?

May 28, 2023, 5:01:54 PM5/28/23

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Moscow,Beijing▂▄▅█████████▅▄▃▂ mushroom cloud, Xi as Putin's stooge when Russia vaporizes Shanghai with its RS-28 Sarmat "Satan II", all because Xi was too dumb to realize you can never trust an insane person

Chris, which of these is Shangai ??? Is it 1/2

> i[0] = 1/1

> i[1] = 1/2

> i[2] = 1/3

> i[3] = 1/4

> i[4] = 1/5

So spamming William, a Sarmat never reaches its exact target but sends shock waves from a blast site.
> i[1] = 1/2

> i[2] = 1/3

> i[3] = 1/4

> i[4] = 1/5

> No.

>

> Not 'to its limit'.

>

> Only TOWARDS its limit.

>

> It never arrives 'at zero'.'

William, infinite spamming nutjob comparing unit fractions of Moscow compared to Beijing. William spamming nutjob is Moscow 0/1 and Beijing fraction is 1/0 ??
>

> Not 'to its limit'.

>

> Only TOWARDS its limit.

>

> It never arrives 'at zero'.'

yan wyck in his daily spam says Beijing was nothing in 1999 and apparently a Russian Sarmat returned Beijing to "nothingness once again".

> Shanghai, Beijing, Shenzhen, Guangzhou, Chongqing, Tianjin, Chengdu, Hangzhou, Nanjing, Wuhan, Xi'An, Suzhou, Harbin, Shenyang, Qingdao, Zhengzhou, Dongguan, Foshan, Dalian, Jinan, Changchun, Hefei

>

> Shenzhen▂▄▅█████████▅▄▃▂ Xi as Putin's stooge when Russia vaporizes Shenzhen with its RS-28 Sarmat "Satan II", all because Xi was too dumb to realize you can never trust an insane person

>

>

>

> Is Pete Olcott in his Halting Problem, halting the vaporization of Wuhan by Putin's Russia???

>

>

> On Thursday, March 30, 2023 at 11:56:20 PM UTC-5, Volney wrote:

> > Botfly of Math and Blowfly of Physics "Putin's stooge"

> >"wasn't bolted down too tight in the first place"

>

> On Friday, September 9, 2022 at 1:16:55 AM UTC-5, Michael Moroney wrote:

> > "Imp of Science"

> >"not one single marble of commonsense in my entire brain"

>

> Moscow█۞█ blackout, knock out Moscow electric power lines█۞█

> & wrote:

> > _And as the Baby Xi grew up from the rice paddies and reeds of Outer

> > Manchuria, stolen by the Naxi and Zani Dictator Putin in Moscow, Xi

> > learned in school in chemical engineering that Taiwan was 1/28 the size

> > of Outer Manchuria, Emperor Qing's homeland, now occupied by homeless Russians drinking vodka, as Putin bombs Ukraine. And the nascent Xi orders

> > 1,000 divisions to the Outer Manchuria border to regain back the stolen

>

>

> > > Why Putin is 2X smarter than Xi as dictators// SCIENCE COUNCIL RULES EARTH, not petty dictators

> > > 2m views

> > >

> > >

> > 2> If Putin pushes nuclear buttons, he drags down China along with Russia into a nuclear ash waste pile, and this means Xi is a inferior junior partner to Putin. Putin will drag down Xi's China, never the reverse.

> > >

> > 2> So, one can look at the present situation on Earth and ask several logical questions about the 2 dictators of Putin's Russia and China's Xi.

> > >

> > > It is little wonder that both Russia and China dictators are combative towards the West. Because dictators never want to give up on power but stay in power all their life long. So they oppose the West because the West has grown up to democracy-- let the people have power, not one single idiot having power all his life time.

> > >

> > > Naturally, Putin will want to keep the Russian people suppressed and have Russia be a second rate government as a dictator. Same goes for China-- they never want to give up power so the people themselves choose their leader.

> > >

> > > But can we find differences in Putin and Xi themselves? Well in the West we call the Chinese inscrutable-- meaning -- little logical commonsense. And is this a valid description?? Yes of course, considering that Russia had stolen the lands of Outer Manchuria, some 28 times larger of a land mass than is Taiwan island. Yet there is Xi, spending so much time on wanting to invade Taiwan, when it is Outer Manchuria and Vladivostok (Haishenwai) that he should be focusing attention upon. And while Putin is distracted with Ukraine, is the time for Xi to recapture Outer Manchuria, the Qing dynasty empire, Qing's Manchurian homeland.

> > >

> > > What does Xi do instead??? He focuses on Taiwan and befriends Russia. Why, at this rate, if Russia takes Inner Manchuria, we can expect Xi and the Chinese Communist Party to become even more loving of Russia for stealing more land of China.

> > >

> > > And there is Xi, whose China has become rich with trading with the West, yet every day, Xi foaming at the mouth in hatred of the West.

> > >

> > > So yes, Putin is 2X smarter as a dictator than is Xi, as if Putin has Xi in his side pocket.

> > >

> > > Is there some scientific explanation as to why Xi is 2X dumber than Putin?? Perhaps, in that China is densely populated and the air pollution over all of China is worse than most countries. That Xi probably has 1/2 of his brain filled with CO and CO2 isomers and lead, and mercury and nitrous oxide and sulfur dioxide from just living in that air polluted hellhole of Beijing. Xi studied chemistry and should know this. Whereas Putin likely detox..s every evening with breathing in pure oxygen at his residence and takes oxygen breathing tanks to office and work. This easily can explain the light-headed reasoning that Xi and his foreign diplomats Wang Yi display, where Putin plays them like a chess game, --- checkmate in 7 moves.

> > >

> > > This explains why Xi hates the West for not stealing any Chinese lands and making China rich in trade, while loving Putin for stealing Outer Manchuria, and proposing having Russia push nuclear buttons, making both Russia and China a nuclear waste site after ICBMs wipe China off the map.

> > >

> > > Xi's brain is full of air pollution toxins from the nasty Chinese air. They still build a new coal fired plant in China every day. The air in China is the worst air in the entire world.

> > >

> > > Why Putin is 2X smarter than Xi as dictators// SCIENCE COUNCIL RULES EARTH, not petty dictators.

>

> > > > 2/1, AP tards:

> > > > > Give Ukraine drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity

> > > >

> > > > 2/2, AP tards:

> > > > > Give Ukraine drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity

> > > >

> > > > 2/3, AP tards:

> > > > > Every Russian missile fired into Ukraine met with a drone from Ukraine knocking out Moscow electric power lines

> > > > >

> > > > > Give Ukraine drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity

> > > >

> > > > 2/4, AP tards:

> > > > > drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity

> > > >

> > > > 2/9 (vacation?), AP tards:

> > > > > drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity

> > > >

> > > > 2/9, AP tards (again):

> > > > > drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity

> > > >

> > > > 2/10, AP tards:

> > > > > drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity

> > > >

> > > > 2/11, AP tards:

> > > > > drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity

> > > >

> > > > 2/12, AP tards:

> > > > > _drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity

> > > >

> > > > 2/12, AP tards again:

> > > > > _drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity

> > > >

> > > > 2/13, AP tards:

> > > > > _drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity

> > > >

> > > > 2/14, AP tards:

> > > > > _drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity

> > > >

> > > > 2/15, AP tards:

> > > > > _drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity

> > > >

> > > > 2/16, AP tards:

> > > > > _drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity

> > > >

> > > > 2/17, AP tards:

> > > > > _drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity

> > > >

> > > > 2/18, AP tards:

> > > > > _drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity

> > > >

> > > > 2/19, AP tards:

> > > > > _drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity

> > > >

> > > > 2/20, AP tards:

> > > >

> > > > > Electricity out Novosibirsk &Volgograd█۞█knock out Moscow electric power lines█۞█

> > > > > _drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity

> > > >

> > > > 2/22, AP tards:

> > > > > Moscow electric blackout█۞█knock out Moscow electric power lines█۞█

> > > > > _drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity

> > > >

> > > > 2/23, AP tards:

> > > > > Moscow electric blackout█۞█knock out Moscow electric power lines█۞█

> > > > > _drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity

> > > >

> > > > 2/24, AP tards:

> > > > > Moscow electric blackout█۞█knock out Moscow electric power lines█۞█

> > > > > _drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity

> > > >

> > > > 2/25, AP tards:

> > > > > Moscow electric blackout█۞█knock out Moscow electric power lines█۞█

> > > > > _drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity

> > > >

> > > > 2/26, AP tards:

May 28, 2023, 5:15:11 PM5/28/23

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On 5/28/2023 2:01 PM, Archimedes Plutonium wrote:

>

>

> Moscow,Beijing▂▄▅█████████▅▄▃▂ mushroom cloud, Xi as Putin's stooge when Russia vaporizes Shanghai with its RS-28 Sarmat "Satan II", all because Xi was too dumb to realize you can never trust an insane person

>

> Chris, which of these is Shangai ??? Is it 1/2

>> i[0] = 1/1

>> i[1] = 1/2

>> i[2] = 1/3

>> i[3] = 1/4

>> i[4] = 1/5

[...]
>

>

> Moscow,Beijing▂▄▅█████████▅▄▃▂ mushroom cloud, Xi as Putin's stooge when Russia vaporizes Shanghai with its RS-28 Sarmat "Satan II", all because Xi was too dumb to realize you can never trust an insane person

>

> Chris, which of these is Shangai ??? Is it 1/2

>> i[0] = 1/1

>> i[1] = 1/2

>> i[2] = 1/3

>> i[3] = 1/4

>> i[4] = 1/5

i[1] = 1/2

May 28, 2023, 5:52:36 PM5/28/23

to

on 5/28/2023, Chris M. Thomasson supposed :

This sequence does not converge.

May 28, 2023, 5:57:26 PM5/28/23

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May 28, 2023, 6:34:35 PM5/28/23

to

On Sunday, May 28, 2023 at 11:52:36 PM UTC+2, FromTheRafters wrote:

> on 5/28/2023, Chris M. Thomasson supposed :

> >

> > 1/1, 1/2, 1/3, ... on its way to its limit [...] zero.
> on 5/28/2023, Chris M. Thomasson supposed :

> >

> >

> This sequence does not converge.

Huh?!
> This sequence does not converge.

The sequence (1/n)_(n e N) does not converge?

0 is not its limit? (i.e. lim_(n -> oo) 1/n =/= 0 ???).

Well... Interesting (sort of).

May 28, 2023, 9:03:47 PM5/28/23

to

Chris M. Thomasson pretended :

Yes, and you can have a sequence converging to any element of the set.

Some sequences will not converge (they might converge in the reals) to

an element of the set of unit fractions. To me, nothing up there says

that this set is not discrete.

Some sequences will not converge (they might converge in the reals) to

an element of the set of unit fractions. To me, nothing up there says

that this set is not discrete.

May 28, 2023, 9:11:53 PM5/28/23

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On Monday, May 29, 2023 at 3:03:47 AM UTC+2, FromTheRafters wrote:

> Chris M. Thomasson pretended :

> > On 5/28/2023 2:50 PM, FromTheRafters wrote:

> >> on 5/28/2023, Chris M. Thomasson supposed:

> >>>

> Chris M. Thomasson pretended :

> > On 5/28/2023 2:50 PM, FromTheRafters wrote:

> >> on 5/28/2023, Chris M. Thomasson supposed:

> >>>

> >>> 1/1, 1/2, 1/3, ... on its way to its limit at zero.

> >>>

> >> This sequence does not converge.

> >

> > It certainly [converges].
> >>>

> >> This sequence does not converge.

> >

> >

> Yes, and <bla>

The sequence does not converge and it converges? Are you doing the Mückenheim her?

May 29, 2023, 5:31:10 AM5/29/23

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Fritz Feldhase used his keyboard to write :

Sure, when you misrepresent (strawman) what I said it sure looks that

way. A sequence of rationals which approaches the value of Pi does not

converge in the rationals because Pi is not a rational number.

Sure, it gets closer and closer to Pi, but Pi is not actually in the

set. Zero is not in the set of unit fractions, so how can a sequence of

unit fractions converge to it in the rationals?

way. A sequence of rationals which approaches the value of Pi does not

converge in the rationals because Pi is not a rational number.

Sure, it gets closer and closer to Pi, but Pi is not actually in the

set. Zero is not in the set of unit fractions, so how can a sequence of

unit fractions converge to it in the rationals?

May 29, 2023, 9:22:35 AM5/29/23

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On Monday, May 29, 2023 at 11:31:10 AM UTC+2, FromTheRafters wrote:

> Zero is not in the set of unit fractions, so how can a sequence of

> unit fractions converge to it in the rationals?

Holy shit! 0 is not a rational number in your book?
> Zero is not in the set of unit fractions, so how can a sequence of

> unit fractions converge to it in the rationals?

Trying to do the Mückenheim?

You are talking nonsense, man.

(Another hint: We usually don't restrict our consideration to the rational numbers when talking about the convergence of a sequence of real numbers. The sequence (1/1, 1/2, 1/3, ...) concerges, and its limit is 0.)

EOD

May 29, 2023, 1:28:17 PM5/29/23

to

Two contradicting results cannot exist in mathematics. The second result is pure logic. The first one is not enforced by logic and avoidable by dark numbers..

Regards, WM

May 29, 2023, 2:09:50 PM5/29/23

to

On Monday, May 29, 2023 at 10:28:17 AM UTC-7, WM wrote:

> NUF(0) = 0. NUF(1) = many. Therefore the unit fractions start between 0 and 1.

Thus our beloved professor reveals himself to be a devotee of the Kalam cosmological argument.
> NUF(0) = 0. NUF(1) = many. Therefore the unit fractions start between 0 and 1.

https://en.wikipedia.org/wiki/Kalam_cosmological_argument

https://en.wikipedia.org/wiki/The_Kal%C4%81m_Cosmological_Argument

May 29, 2023, 2:14:01 PM5/29/23

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Fritz Feldhase expressed precisely :

book.

> On Monday, May 29, 2023 at 11:31:10 AM UTC+2, FromTheRafters wrote:

>

>> Zero is not in the set of unit fractions, so how can a sequence of

>> unit fractions converge to it in the rationals?

>

> Holy shit! 0 is not a rational number in your book?

It is not a unit fraction in my book. It is not a positive real in my
>

>> Zero is not in the set of unit fractions, so how can a sequence of

>> unit fractions converge to it in the rationals?

>

> Holy shit! 0 is not a rational number in your book?

book.

May 29, 2023, 3:50:45 PM5/29/23

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On Monday, May 29, 2023 at 2:28:17 PM UTC-3, WM wrote:

> William schrieb am Sonntag, 28. Mai 2023 um 17:44:28 UTC+2:

> > we have

> > ∀n ∈ ℕ: 1/n - 1/(n+1) = 1/(n(n+1)) > 0 .

> > note the universal quantifier. This holds for every unit fraction.

>...and avoidable by dark numbers..
> William schrieb am Sonntag, 28. Mai 2023 um 17:44:28 UTC+2:

> > we have

> > ∀n ∈ ℕ: 1/n - 1/(n+1) = 1/(n(n+1)) > 0 .

> > note the universal quantifier. This holds for every unit fraction.

Hence, unit fractions cannot be "dark numbers".

May 29, 2023, 3:52:34 PM5/29/23

to

On Monday, May 29, 2023 at 3:14:01 PM UTC-3, FromTheRafters wrote:

> Fritz Feldhase expressed precisely :

> > On Monday, May 29, 2023 at 11:31:10 AM UTC+2, FromTheRafters wrote:

> >

> >> Zero is not in the set of unit fractions, so how can a sequence of

> >> unit fractions converge to it in the rationals?

> >

> > Holy shit! 0 is not a rational number in your book?

> It is not a unit fraction in my book. It is not a positive real in my

> book.

https://xkcd.com/169/
> Fritz Feldhase expressed precisely :

> > On Monday, May 29, 2023 at 11:31:10 AM UTC+2, FromTheRafters wrote:

> >

> >> Zero is not in the set of unit fractions, so how can a sequence of

> >> unit fractions converge to it in the rationals?

> >

> > Holy shit! 0 is not a rational number in your book?

> It is not a unit fraction in my book. It is not a positive real in my

> book.

May 29, 2023, 4:20:08 PM5/29/23

to

On 5/29/2023 2:29 AM, FromTheRafters wrote:

> Fritz Feldhase used his keyboard to write :

>> On Monday, May 29, 2023 at 3:03:47 AM UTC+2, FromTheRafters wrote:

>>> Chris M. Thomasson pretended :

>>>> On 5/28/2023 2:50 PM, FromTheRafters wrote:

>>>>> on 5/28/2023, Chris M. Thomasson supposed:

>>>>>>

>>>>>> 1/1, 1/2, 1/3, ... on its way to its limit at zero.

>>>>> This sequence does not converge.

>>>>

>>>> It certainly [converges].

>>>>

>>> Yes, and <bla>

>>

>> The sequence does not converge and it converges? Are you doing the

>> Mückenheim her?

>

> Sure, when you misrepresent (strawman) what I said it sure looks that

> way. A sequence of rationals which approaches the value of Pi does not

> converge in the rationals because Pi is not a rational number.

The iterates get closer and closer to zero... Their limit.
> Fritz Feldhase used his keyboard to write :

>> On Monday, May 29, 2023 at 3:03:47 AM UTC+2, FromTheRafters wrote:

>>> Chris M. Thomasson pretended :

>>>> On 5/28/2023 2:50 PM, FromTheRafters wrote:

>>>>> on 5/28/2023, Chris M. Thomasson supposed:

>>>>>>

>>>>>> 1/1, 1/2, 1/3, ... on its way to its limit at zero.

>>>>> This sequence does not converge.

>>>>

>>>> It certainly [converges].

>>>>

>>> Yes, and <bla>

>>

>> The sequence does not converge and it converges? Are you doing the

>> Mückenheim her?

>

> Sure, when you misrepresent (strawman) what I said it sure looks that

> way. A sequence of rationals which approaches the value of Pi does not

> converge in the rationals because Pi is not a rational number.

May 29, 2023, 4:35:57 PM5/29/23

to

"Chris M. Thomasson" <chris.m.t...@gmail.com> writes:

>>>>>> on 5/28/2023, Chris M. Thomasson supposed:

>>>>>>> 1/1, 1/2, 1/3, ... on its way to its limit at zero.

>>>>>> on 5/28/2023, Chris M. Thomasson supposed:

>>>>>>> 1/1, 1/2, 1/3, ... on its way to its limit at zero.

> The iterates get closer and closer to zero... Their limit.

You know there are books and online courses/tutorials from which you
could learn what a limit is? Getting "closer and closer to zero" does

not mean that zero is the limit. Don't get me wrong -- zero /is/ the

limit in this case, but if you were a student of mine I'd invite you to

define a sequence that gets forever "closer and closer to zero" but

which does /not/ converge to a limit of zero.

--

Ben.

May 29, 2023, 4:40:13 PM5/29/23

to

Regards, WM

May 29, 2023, 4:43:19 PM5/29/23

to

On Monday, May 29, 2023 at 8:14:01 PM UTC+2, FromTheRafters wrote:

> Fritz Feldhase expressed precisely :

> > On Monday, May 29, 2023 at 11:31:10 AM UTC+2, FromTheRafters wrote:

> >

> >> Zero is not in the set of unit fractions, so how can a sequence of

> >> unit fractions converge to it in the rationals?

> >

> > Holy shit! 0 is not a rational number in your book?

> It is not a unit fraction in my book. It is not a positive real in my

> book.

??? Ok, you are an idiot, I see.
> Fritz Feldhase expressed precisely :

> > On Monday, May 29, 2023 at 11:31:10 AM UTC+2, FromTheRafters wrote:

> >

> >> Zero is not in the set of unit fractions, so how can a sequence of

> >> unit fractions converge to it in the rationals?

> >

> > Holy shit! 0 is not a rational number in your book?

> It is not a unit fraction in my book. It is not a positive real in my

> book.

*plonk*

> > Trying to do the Mückenheim?

> >

> > You are talking nonsense, man.

> >

> > (Another hint: We usually don't restrict our consideration to the rational

> > numbers when talking about the convergence of a sequence of real numbers. The

> > sequence (1/1, 1/2, 1/3, ...) concerges, and its limit is 0.)

> >

> > EOD

May 29, 2023, 4:46:47 PM5/29/23

to

1/2 + 1/3 + 1/4 + 1/5 ... ?

The terms get closer to zero, but the sum always gets bigger?

i[0] = 1/2

i[1] = i[0] + 1/3

i[2] = i[1] + 1/4

i[3] = i[2] + 1/5

...

?

May 29, 2023, 4:57:23 PM5/29/23

to

fraction, 1/2, 1/3, ..., is their journey down to getting closer and

closer to their limit at 0/1...

May 29, 2023, 5:01:24 PM5/29/23

to

"Chris M. Thomasson" <chris.m.t...@gmail.com> writes:

> On 5/29/2023 1:35 PM, Ben Bacarisse wrote:

>> "Chris M. Thomasson" <chris.m.t...@gmail.com> writes:

>>

>>>>>>>> on 5/28/2023, Chris M. Thomasson supposed:

>>>>>>>>> 1/1, 1/2, 1/3, ... on its way to its limit at zero.

>>

>>> The iterates get closer and closer to zero... Their limit.

>> You know there are books and online courses/tutorials from which you

>> could learn what a limit is? Getting "closer and closer to zero" does

>> not mean that zero is the limit. Don't get me wrong -- zero /is/ the

>> limit in this case, but if you were a student of mine I'd invite you to

>> define a sequence that gets forever "closer and closer to zero" but

>> which does /not/ converge to a limit of zero.

>>

>

> Humm... Perhaps, something like:

>

> 1/2 + 1/3 + 1/4 + 1/5 ... ?

>

> The terms get closer to zero, but the sum always gets bigger?

No. I meant literally what I said. There no games or word trickery
> On 5/29/2023 1:35 PM, Ben Bacarisse wrote:

>> "Chris M. Thomasson" <chris.m.t...@gmail.com> writes:

>>

>>>>>>>> on 5/28/2023, Chris M. Thomasson supposed:

>>>>>>>>> 1/1, 1/2, 1/3, ... on its way to its limit at zero.

>>

>>> The iterates get closer and closer to zero... Their limit.

>> You know there are books and online courses/tutorials from which you

>> could learn what a limit is? Getting "closer and closer to zero" does

>> not mean that zero is the limit. Don't get me wrong -- zero /is/ the

>> limit in this case, but if you were a student of mine I'd invite you to

>> define a sequence that gets forever "closer and closer to zero" but

>> which does /not/ converge to a limit of zero.

>>

>

> Humm... Perhaps, something like:

>

> 1/2 + 1/3 + 1/4 + 1/5 ... ?

>

> The terms get closer to zero, but the sum always gets bigger?

going on. Find a sequence whose terms get closer and closer to zero but

which does not converge to zero. It's much simpler than you think.

.

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Hint: for the terms of s(n) to get "closer and closer to zero" all we

need is that

|s(n+1) - 0| < |s(n) - 0|

--

Ben.

May 29, 2023, 5:06:57 PM5/29/23

to

On 5/29/2023 1:59 PM, Ben Bacarisse wrote:

> "Chris M. Thomasson" <chris.m.t...@gmail.com> writes:

>

>> On 5/29/2023 1:35 PM, Ben Bacarisse wrote:

>>> "Chris M. Thomasson" <chris.m.t...@gmail.com> writes:

>>>

>>>>>>>>> on 5/28/2023, Chris M. Thomasson supposed:

>>>>>>>>>> 1/1, 1/2, 1/3, ... on its way to its limit at zero.

>>>

>>>> The iterates get closer and closer to zero... Their limit.

>>> You know there are books and online courses/tutorials from which you

>>> could learn what a limit is? Getting "closer and closer to zero" does

>>> not mean that zero is the limit. Don't get me wrong -- zero /is/ the

>>> limit in this case, but if you were a student of mine I'd invite you to

>>> define a sequence that gets forever "closer and closer to zero" but

>>> which does /not/ converge to a limit of zero.

>>>

>>

>> Humm... Perhaps, something like:

>>

>> 1/2 + 1/3 + 1/4 + 1/5 ... ?

>>

>> The terms get closer to zero, but the sum always gets bigger?

>

> No. I meant literally what I said. There no games or word trickery

> going on. Find a sequence whose terms get closer and closer to zero but

> which does not converge to zero. It's much simpler than you think.

[...]
> "Chris M. Thomasson" <chris.m.t...@gmail.com> writes:

>

>> On 5/29/2023 1:35 PM, Ben Bacarisse wrote:

>>> "Chris M. Thomasson" <chris.m.t...@gmail.com> writes:

>>>

>>>>>>>>> on 5/28/2023, Chris M. Thomasson supposed:

>>>>>>>>>> 1/1, 1/2, 1/3, ... on its way to its limit at zero.

>>>

>>>> The iterates get closer and closer to zero... Their limit.

>>> You know there are books and online courses/tutorials from which you

>>> could learn what a limit is? Getting "closer and closer to zero" does

>>> not mean that zero is the limit. Don't get me wrong -- zero /is/ the

>>> limit in this case, but if you were a student of mine I'd invite you to

>>> define a sequence that gets forever "closer and closer to zero" but

>>> which does /not/ converge to a limit of zero.

>>>

>>

>> Humm... Perhaps, something like:

>>

>> 1/2 + 1/3 + 1/4 + 1/5 ... ?

>>

>> The terms get closer to zero, but the sum always gets bigger?

>

> No. I meant literally what I said. There no games or word trickery

> going on. Find a sequence whose terms get closer and closer to zero but

> which does not converge to zero. It's much simpler than you think.

Humm... For some reason I am thinking about alternating the signs of the

terms?

1/1, -1/2, 1/3, -1/4, 1/5, -1/6, ?

Damn, can't be right. I need to work on some other stuff right now, but

I will get back to you. Thanks Ben! :^)

May 29, 2023, 5:09:27 PM5/29/23

to

1, -1, 1, -1

Not sure exactly why I thought of it.

May 29, 2023, 5:18:34 PM5/29/23

to

On 5/29/2023 2:04 PM, Chris M. Thomasson wrote:

so damn. Sorry.

May 29, 2023, 6:33:47 PM5/29/23

to

picked one meaning for getting closer and closer to zero, and you can't

see any other.

--

Ben.

May 29, 2023, 6:38:30 PM5/29/23

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it is true for every unit fraction that there is a second unit fraction between it and zero

May 29, 2023, 7:29:51 PM5/29/23

to

i[0] = 10

i[1] = i[0] / 2

i[2] = i[1] / 2

i[3] = i[2] / 2

i[4] = i[3] / 2

i[5] = i[4] / 2

...

It gets closer and closer to zero. But, its not what you asked for.

May 29, 2023, 8:57:57 PM5/29/23

to

closer and closer to zero, forever? What is the limit?

--

Ben.

May 29, 2023, 9:49:02 PM5/29/23

to

WM <askas...@gmail.com> writes:

(AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des

Unendlichen" at Hochschule Augsburg.)

> William schrieb am Sonntag, 28. Mai 2023 um 17:44:28 UTC+2:

>> we have

>> ∀n ∈ ℕ: 1/n - 1/(n+1) = 1/(n(n+1)) > 0 .

>> note the universal quantifier. This holds for every unit fraction. Thus for every unit fraction (unrevealed or invisible or ...) there is another unit fraction closer to zero (the gap is finite.),

>> Thus the is no first unit fraction.

>

> This is contradicted by the following: NUF(0) = 0. NUF(1) =

> many. Therefore the unit fractions start between 0 and 1. Never more

> than one occupy a point. This enforces a first one. No way to circ

> umvent this conclusion.

Oh dear. Then, according to you, there is another error in your book.

You give the set of rationals B = {x > sqrt(2)} and say that it has no

smallest element. Define NR(z) as the number of elements of B less than

z. Now NR(sqrt(2)) = 0 and NR(2) = many so the elements of your set B

start between sqrt(2) and 2. Never more than one occupy a point. This

enforces a first one, yet you say that B has no first element.

--

Ben.

(AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des

Unendlichen" at Hochschule Augsburg.)

> William schrieb am Sonntag, 28. Mai 2023 um 17:44:28 UTC+2:

>> we have

>> ∀n ∈ ℕ: 1/n - 1/(n+1) = 1/(n(n+1)) > 0 .

>> note the universal quantifier. This holds for every unit fraction. Thus for every unit fraction (unrevealed or invisible or ...) there is another unit fraction closer to zero (the gap is finite.),

>> Thus the is no first unit fraction.

>

> This is contradicted by the following: NUF(0) = 0. NUF(1) =

> many. Therefore the unit fractions start between 0 and 1. Never more

> than one occupy a point. This enforces a first one. No way to circ

> umvent this conclusion.

You give the set of rationals B = {x > sqrt(2)} and say that it has no

smallest element. Define NR(z) as the number of elements of B less than

z. Now NR(sqrt(2)) = 0 and NR(2) = many so the elements of your set B

start between sqrt(2) and 2. Never more than one occupy a point. This

enforces a first one, yet you say that B has no first element.

--

Ben.

May 30, 2023, 11:07:15 AM5/30/23

to

Regards, WM

May 30, 2023, 11:13:35 AM5/30/23

to

Ben Bacarisse schrieb am Dienstag, 30. Mai 2023 um 03:49:02 UTC+2:

> WM <askas...@gmail.com> writes:

> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des

> Unendlichen" at Hochschule Augsburg.)

> > William schrieb am Sonntag, 28. Mai 2023 um 17:44:28 UTC+2:

> >> we have

> >> ∀n ∈ ℕ: 1/n - 1/(n+1) = 1/(n(n+1)) > 0 .

> >> note the universal quantifier. This holds for every unit fraction. Thus for every unit fraction (unrevealed or invisible or ...) there is another unit fraction closer to zero (the gap is finite.),

> >> Thus the is no first unit fraction.

> >

> > This is contradicted by the following: NUF(0) = 0. NUF(1) =

> > many. Therefore the unit fractions start between 0 and 1. Never more

> > than one occupy a point. This enforces a first one. No way to circ

> > umvent this conclusion.

> Oh dear. Then, according to you, there is another error in your book.

> You give the set of rationals B = {x > sqrt(2)} and say that it has no

> smallest element.

In my book there are no dark numbers.
> WM <askas...@gmail.com> writes:

> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des

> Unendlichen" at Hochschule Augsburg.)

> > William schrieb am Sonntag, 28. Mai 2023 um 17:44:28 UTC+2:

> >> we have

> >> ∀n ∈ ℕ: 1/n - 1/(n+1) = 1/(n(n+1)) > 0 .

> >> note the universal quantifier. This holds for every unit fraction. Thus for every unit fraction (unrevealed or invisible or ...) there is another unit fraction closer to zero (the gap is finite.),

> >> Thus the is no first unit fraction.

> >

> > This is contradicted by the following: NUF(0) = 0. NUF(1) =

> > many. Therefore the unit fractions start between 0 and 1. Never more

> > than one occupy a point. This enforces a first one. No way to circ

> > umvent this conclusion.

> Oh dear. Then, according to you, there is another error in your book.

> You give the set of rationals B = {x > sqrt(2)} and say that it has no

> smallest element.

> Define NR(z) as the number of elements of B less than

> z. Now NR(sqrt(2)) = 0 and NR(2) = many so the elements of your set B

> start between sqrt(2) and 2. Never more than one occupy a point. This

> enforces a first one, yet you say that B has no first element.

Regards, WM

May 30, 2023, 12:13:03 PM5/30/23

to

On Tuesday, 30 May 2023 at 12:07:15 UTC-3, WM wrote:

[...]

[...]

> My case is this: If the number of unit fractions is 0 at 0, then it must have decreased before.

Holy shit! Even this you manage to get wrong!!!
May 30, 2023, 12:28:54 PM5/30/23

to

On Tuesday, May 30, 2023 at 5:07:15 PM UTC+2, WM wrote:

> If the number of unit fractions [smaller than x] is 0 at x = 0, then it must have decreased before. That is dictated by pure logic.

There are infinitely many unit fractions in (0, 1]. But for each and every real number x > 0, there are only finitely many unit fractions >= x. HENCE for each and every real number x > 0, there are infinitely many unit fractions smaller than x.

That is dictated by classical mathematics (which is based on classical logic and set theory).

See: https://en.wikipedia.org/wiki/Classical_mathematics

> If the number of unit fractions [smaller than x] is 0 at x = 0, then it must have decreased before. That is dictated by pure logic.

There are infinitely many unit fractions in (0, 1]. But for each and every real number x > 0, there are only finitely many unit fractions >= x. HENCE for each and every real number x > 0, there are infinitely many unit fractions smaller than x.

That is dictated by classical mathematics (which is based on classical logic and set theory).

See: https://en.wikipedia.org/wiki/Classical_mathematics

May 30, 2023, 1:21:49 PM5/30/23

to

May 30, 2023, 1:33:53 PM5/30/23

to

Fritz Feldhase schrieb am Dienstag, 30. Mai 2023 um 18:28:54 UTC+2:

> On Tuesday, May 30, 2023 at 5:07:15 PM UTC+2, WM wrote:

>

> > If the number of unit fractions [smaller than x] is 0 at x = 0, then it must have decreased before. That is dictated by pure logic.

>

> There are infinitely many unit fractions in (0, 1]. But for each and every real number x > 0, there are only finitely many unit fractions >= x. HENCE for each and every real number x > 0, there are infinitely many unit fractions smaller than x.

Infinite means ordered and without visible end. Yes, ∀x ∈ (eps, 1]: SBZ(x) = ℵo.
> On Tuesday, May 30, 2023 at 5:07:15 PM UTC+2, WM wrote:

>

> > If the number of unit fractions [smaller than x] is 0 at x = 0, then it must have decreased before. That is dictated by pure logic.

>

> There are infinitely many unit fractions in (0, 1]. But for each and every real number x > 0, there are only finitely many unit fractions >= x. HENCE for each and every real number x > 0, there are infinitely many unit fractions smaller than x.

>

> That is dictated by classical mathematics (which is based on classical logic and set theory).

It contradicts logic and therefore has to be disposed of. When different points are passed such that all have been passed at 0, then a last one before 0 has been passed. That is the more obvious since all have positive internal distances.
> That is dictated by classical mathematics (which is based on classical logic and set theory).

Regards, WM

May 30, 2023, 2:03:48 PM5/30/23

to

On Tuesday, May 30, 2023 at 7:33:53 PM UTC+2, WM wrote:

> Fritz Feldhase schrieb am Dienstag, 30. Mai 2023 um 18:28:54 UTC+2:

> >

> > There are infinitely many unit fractions in (0, 1]. But for each and every real number x > 0, there are only finitely many unit fractions >= x. HENCE for each and every real number x > 0, there are infinitely many unit fractions smaller than x.

> >

> Infinite means ordered

No, /infinite/ does not "mean ordered".
> Fritz Feldhase schrieb am Dienstag, 30. Mai 2023 um 18:28:54 UTC+2:

> >

> > There are infinitely many unit fractions in (0, 1]. But for each and every real number x > 0, there are only finitely many unit fractions >= x. HENCE for each and every real number x > 0, there are infinitely many unit fractions smaller than x.

> >

> Infinite means ordered

> and without visible end.

Fascinating.

Please define *visible*. Then proof your claim.

> Yes, ∀x ∈ (eps, 1]: SBZ(x) = ℵo.

Without "specifying" eps, this is just nonsense.

On the other hand, with eps = 0, we get: ∀x ∈ (0, 1]: SBZ(x) = ℵo (which is indeed true).

May 30, 2023, 3:39:24 PM5/30/23

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one? Is that a wrong way of thinking? Thanks, Ben. :^)

May 30, 2023, 3:42:44 PM5/30/23

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from negative to zero.

May 30, 2023, 3:46:03 PM5/30/23

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(1/1 + 1), (1/2 + 1), (1/3 + 1), (1/4 + 1)...

That should get closer and closer to one simply because I offset it by one?

May 30, 2023, 3:53:33 PM5/30/23

to

by anything I want to and that offset becomes the limit? Is that kook

shit, Ben? Any sequence that limits out on zero can be offset such that

it limits out on another number? Fair enough, or kooky! ;^o

May 30, 2023, 3:56:30 PM5/30/23

to

On 5/30/2023 8:07 AM, WM wrote:

> William schrieb am Dienstag, 30. Mai 2023 um 00:38:30 UTC+2:

>> On Monday, May 29, 2023 at 5:40:13 PM UTC-3, WM wrote:

>>> William schrieb am Montag, 29. Mai 2023 um 21:50:45 UTC+2:

>>>> On Monday, May 29, 2023 at 2:28:17 PM UTC-3, WM wrote:

>>>>> William schrieb am Sonntag, 28. Mai 2023 um 17:44:28 UTC+2:

>>>>>> we have

>>>>>> ∀n ∈ ℕ: 1/n - 1/(n+1) = 1/(n(n+1)) > 0 .

>>>>>> note the universal quantifier. This holds for every unit fraction.

>>>>> ...and avoidable by dark numbers..

>>>> Hence, unit fractions cannot be "dark numbers".

>>> Wrong conclusion. The unit fractions start between 0 and 1. Never more than one occupy a point. This enforces a first one.

>> And since this "fist one" is a unit fraction, anything that is true for all unit fractions must be true for this putative "fist one". In particular

>> it is true for every unit fraction that there is a second unit fraction between it and zero

>

> My case is this: If the number of unit fractions is 0 at 0,

Unit fractions go from one to zero, although they will never equal zero.
> William schrieb am Dienstag, 30. Mai 2023 um 00:38:30 UTC+2:

>> On Monday, May 29, 2023 at 5:40:13 PM UTC-3, WM wrote:

>>> William schrieb am Montag, 29. Mai 2023 um 21:50:45 UTC+2:

>>>> On Monday, May 29, 2023 at 2:28:17 PM UTC-3, WM wrote:

>>>>> William schrieb am Sonntag, 28. Mai 2023 um 17:44:28 UTC+2:

>>>>>> we have

>>>>>> ∀n ∈ ℕ: 1/n - 1/(n+1) = 1/(n(n+1)) > 0 .

>>>>>> note the universal quantifier. This holds for every unit fraction.

>>>>> ...and avoidable by dark numbers..

>>>> Hence, unit fractions cannot be "dark numbers".

>>> Wrong conclusion. The unit fractions start between 0 and 1. Never more than one occupy a point. This enforces a first one.

>> And since this "fist one" is a unit fraction, anything that is true for all unit fractions must be true for this putative "fist one". In particular

>> it is true for every unit fraction that there is a second unit fraction between it and zero

>

> My case is this: If the number of unit fractions is 0 at 0,

You seem to be going the other way around from zero to one. What about:

(1 - 1/1), (1 - 1/2), (1 - 1/3), (1 - 1/4), ...

Now, the go from zero to one, where one is their limit? Fair enough?

May 30, 2023, 3:58:17 PM5/30/23

to

On 5/30/2023 12:56 PM, Chris M. Thomasson wrote:

> On 5/30/2023 8:07 AM, WM wrote:

>> William schrieb am Dienstag, 30. Mai 2023 um 00:38:30 UTC+2:

>>> On Monday, May 29, 2023 at 5:40:13 PM UTC-3, WM wrote:

>>>> William schrieb am Montag, 29. Mai 2023 um 21:50:45 UTC+2:

>>>>> On Monday, May 29, 2023 at 2:28:17 PM UTC-3, WM wrote:

>>>>>> William schrieb am Sonntag, 28. Mai 2023 um 17:44:28 UTC+2:

>>>>>>> we have

>>>>>>> ∀n ∈ ℕ: 1/n - 1/(n+1) = 1/(n(n+1)) > 0 .

>>>>>>> note the universal quantifier. This holds for every unit fraction.

>>>>>> ...and avoidable by dark numbers..

>>>>> Hence, unit fractions cannot be "dark numbers".

>>>> Wrong conclusion. The unit fractions start between 0 and 1. Never

>>>> more than one occupy a point. This enforces a first one.

>>> And since this "fist one" is a unit fraction, anything that is true

>>> for all unit fractions must be true for this putative "fist one". In

>>> particular

>>> it is true for every unit fraction that there is a second unit

>>> fraction between it and zero

>>

>> My case is this: If the number of unit fractions is 0 at 0,

>

> Unit fractions go from one to zero, although they will never equal zero.

>

> You seem to be going the other way around from zero to one. What about:

>

> (1 - 1/1), (1 - 1/2), (1 - 1/3), (1 - 1/4), ...

>

> Now, the go from zero to one, where one is their limit? Fair enough?

Thanks Ben. :^)
> On 5/30/2023 8:07 AM, WM wrote:

>> William schrieb am Dienstag, 30. Mai 2023 um 00:38:30 UTC+2:

>>> On Monday, May 29, 2023 at 5:40:13 PM UTC-3, WM wrote:

>>>> William schrieb am Montag, 29. Mai 2023 um 21:50:45 UTC+2:

>>>>> On Monday, May 29, 2023 at 2:28:17 PM UTC-3, WM wrote:

>>>>>> William schrieb am Sonntag, 28. Mai 2023 um 17:44:28 UTC+2:

>>>>>>> we have

>>>>>>> ∀n ∈ ℕ: 1/n - 1/(n+1) = 1/(n(n+1)) > 0 .

>>>>>>> note the universal quantifier. This holds for every unit fraction.

>>>>>> ...and avoidable by dark numbers..

>>>>> Hence, unit fractions cannot be "dark numbers".

>>>> Wrong conclusion. The unit fractions start between 0 and 1. Never

>>>> more than one occupy a point. This enforces a first one.

>>> And since this "fist one" is a unit fraction, anything that is true

>>> for all unit fractions must be true for this putative "fist one". In

>>> particular

>>> it is true for every unit fraction that there is a second unit

>>> fraction between it and zero

>>

>> My case is this: If the number of unit fractions is 0 at 0,

>

> Unit fractions go from one to zero, although they will never equal zero.

>

> You seem to be going the other way around from zero to one. What about:

>

> (1 - 1/1), (1 - 1/2), (1 - 1/3), (1 - 1/4), ...

>

> Now, the go from zero to one, where one is their limit? Fair enough?

May 30, 2023, 7:05:14 PM5/30/23

to

also get closer and closer to 1/2. And they get closer and closer to

zero. We usually measure closeness by distance, and the most natural

notion of the distance between two numbers a and b is the absolute

difference |a-b|. |2-0| = 2 but |3/2 - 0| = 3/2 because 3/2 is closer

to zero than 2 is. Each member of the 1+1/n sequence is closer to zero

than all the previous ones.

A limit is (very loosely) a number that some sequence gets /arbitrarily/

close to (and, eventually, stays close to). That is not the same as

simply getting closer and closer to some number.

>> How about,

>> (1/1 + 1), (1/2 + 1), (1/3 + 1), (1/4 + 1)...

>> That should get closer and closer to one simply because I offset it by

>> one?

>

> Since the unit fractions get closer and closer to zero, I can offset it by

> anything I want to and that offset becomes the limit? Is that kook shit,

> Ben? Any sequence that limits out on zero can be offset such that it limits

> out on another number? Fair enough, or kooky! ;^o

will give you a whole host of similar theorems.

--

Ben.

May 31, 2023, 10:30:23 AM5/31/23

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Fritz Feldhase schrieb am Dienstag, 30. Mai 2023 um 20:03:48 UTC+2:

> On Tuesday, May 30, 2023 at 7:33:53 PM UTC+2, WM wrote:

> > Yes, ∀x ∈ (eps, 1]: SBZ(x) = ℵo.

>

> On Tuesday, May 30, 2023 at 7:33:53 PM UTC+2, WM wrote:

> > Yes, ∀x ∈ (eps, 1]: SBZ(x) = ℵo.

>

> Without "specifying" eps, this is just nonsense.

Every eps > 0 that can be chosen can be inserted here.
∀x ∈ (eps, 1]: SBZ(x) = ℵo.

>

>

> On the other hand, with eps = 0, we get: ∀x ∈ (0, 1]: SBZ(x) = ℵo (which is indeed true).

It is wrong as wrong can be. Starting from zero (who could forbid it?) we see that there must be a first unit fraction, because two or more are separated by gaps.
Regards, WM

May 31, 2023, 10:32:23 AM5/31/23