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Message from discussion Polishing up II

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More options Jun 25 2012, 7:03 am
Newsgroups: alt.algebra.help
From: William Elliot <ma...@panix.com>
Date: Mon, 25 Jun 2012 04:03:53 -0700
Local: Mon, Jun 25 2012 7:03 am
Subject: Re: Polishing up II

On Mon, 25 Jun 2012, Jack wrote:
> Once again I have got this issue of wanting to say two things in one (and
> ideally I want to be able to reference the assertion, all in one), namely
> that f(x)/g(x) is  strictly decreasing and approaches 1.

f(x) strictly decreases to 1 as x -> oo.

> Currently I am writing
> 'f(x)/g(x) ~ 1, the quotient being strictly decreasing'

I suppose.  So when defining f(x) ~ r, don't be glib,
spell it all out.  f is a monotonically decreasing function
with lim(x->oo) f(x) = r.

> or
> 'lim_(x-->oo) f(x)/g(x) = 1
> with f(x)/g(x) >1'.

No, that's saying something different.

> I don't know if this is the best phrasing; I have read that the notation I
> am using here is known as asymptotic notation but my functions are
> monotonic -- the curves they produce are smooth. Any views?

Yes, asymptotic and monotone are different.

> Also, I have changed the sentence <<For x<y in mathbb{N}, [x,y] = {n n
> mathbb{N} : x <= n <= y} >> given to me by Paul, to <<all intervals are
> be taken to be intervals of integers>>. I wonder whether this is OK,
> given that Paul's version might be seen to be doing something more,
> namely indicating that the square brackets denote endpoints. Is the
> open/-closed-bounded notation using round and square brackets
> respectively, sufficiently standard that I don't need to worry?

{n n > mathbb{N} : x <= n <= y}

is a typo and skip the TeX.

In Europe, they use ]a,b[ for (a,b).

When restriced to integers only
one is needed for (a,b) = [a-1, b+1].