The example given can be looked at /intuitively/ as "the staircases
shrinking until they become the line". But that is /not/ a mathematical
statement; it's "what it looks like is happening" - it is a description
of our /intuition/..

The actual mathematical statement is this:

First, let C = (C_1, C_2, C_3, ...,C_n, ...) be a sequence of sets of
points. (You have previously agreed that lines, curves, disks, etc. can
be considered as simply sets of points).

Call a point p a "limit point" of the sequence C if there exists a
sequence (p_1, p_2, p_3, ..., p_n, ...) such that, for each k, p_k is a
point in C_k; and such that the limit of (p_1, p_2, p_3, ..., p_n, ...)
is p.

So for example, if the curves {C_n} are the staircases, and p is the
point (1/2, 1/2), then let p_k be the point of intersection between the
line x=y and the staircase C_k. Then the seqeunce ((1,1), (1/2,1/2),
(3/4,3/4), (1/2,1/2),....) has limit p =(1/2,1/2); and so p is a limit
point of the sequence C.

When I say "limit of the sequence (p_1, p_2, ...)", I mean the same old
"garden variety" limit we are used to (see below for details). It's
worth noting here that this definition relies on the usual euclidean
metric when we say "approaches".

So to say "p is a limit point of C" is /not/ saying that "the
staircases in C disappear" or "become infinitesimally small". The
elements in the sequence C are always what they are : in our case,
staircases, with an increasingly small, but always finite, step size.
There is /no/ element of C which is /not/ an actual staircase.

"p is a limit point of C" is instead a statement that the curves in C
get arbitrarily close to the point p. That's all it claims; just like
the statement "p is the limit of the sequence (p_1, p_2, ..., p_n,
...)" doesn't claim that there is some p_k such that p_k = p; i.e.,
that somehow the points in the sequence "become" p.

With me so far? Is it clear what it means to say that "p is a limit
point of C"?

Now, define "the limit of C" as the set of /all/ points p such that p
is a limit point of C. Again, this is not saying "therefore the
staircases get infinitely small", or "become infinitesimal" or any such
thing. It is a set of points; each point having the property: the
curves in C get arbitrarily close to that point.

What do we have when we're done? A set of points, and that's it. No
"direction-points", no "infinitesimal lines", nothing like that: just
plain old, vanilla points.

You can think of the limit in this case a function: pass in a sequence
of sets of points, and you get out the single set of points that
satisfy the above definition.

Notice that this definition of limit can also "return" the empty set.
Suppose D is the diagonal of the unit square, and X is the line segment
between the points (5,5) and (6,6). Then the sequence (D, X, D, X,
....) has limit the empty set: there are no points which meet the
definition "p is a limit point of that sequence".

Also notice that this definition of limit corresponds to our usual
intuitions about the limit of a set of curves: this definition
identifies the circle as the limit of polygons inscribed in a circle,
as well as the "usual" limit of the polygonal curve which we use to
find the lengh of an arbitrary continuous function f in calculus.

At one point you say "the line is not just a set of points", and at
another point you say "I accept that the line is just a set of points".
Which is it?

<snip>

Similarly, I don't claim that the staircase "becomes" the diagonal; I
claim that the /sequence/ of staircases /approaches/ the diagonal
arbitrarily closely.

<snip>

Does the formula y = 1-x define the sequence of staircases? Of course
not! It's the other way around: the limit of the sequence of staircases
is defined to be the diagonal, which is equivalent to (a segment of)
the line y = 1 - x.

By /the definition/.

(1) I use it implicitly when I use the euclidean metric to say "(p_1,
p_2, ..., p_n, ...) has limit p".

(2) I use it when I state that for every staircase C_n, that staircase
has length 2.

The limit of (the length of the nth staircase)

is not the same as

The length of (the limit of the staircases)

is /exactly/ the point I'm making - the insistence that these two
values must be equal is Premise B, not Premise A. Once we have an
actual /definition/ for the limit of the staircases, we can see that
Premise B is false in general.

<snip>

Does it ever get to be less than -1? Doesn't look like it to me: and
that makes "-1" a lower bound.

In fact, 0 is also a lower bound; but any number greater than 0 is not
a lower bound; so we could call 0 a "greatest lower bound".

But that is /still/ not sufficient to say "0 is the limit of this
sequence". 0 is /also/ a greatest lower bound of the sequence
(0,1,0,1,...). Bu this sequence has no limit. So you are /missing/
something in your understanding of "the limit".

If the sequence is (0, pi, 0, 2*pi, 0, 3*pi, ...) is the limit pi? It
is certainly the case for any (Hint: NON-ZERO POSITIVE REAL) finite x,
there is a finite n (n=2, a_2 = pi) such that a_n - pi = a_2 - pi = pi
- pi = 0 < x. Correct? So is the limit of that sequence pi?

So, now what do you think it really means to claim that the sequence
(a_1,a_2, ..., a_n, ...) = pi?

<snip>

Consider the function f defined as: f(x) = 0 if x <1, and f(x)=1 if
x>=1. Do you accept that this is a valid function, just as "size" or
"length" are functions?

If the answer is yes, do you agree that g(n) = f((2^n-1)/2^n) is also a
valid function, where f is defined as above?

Do you agree that g(0)=0? Do you agree that if g(n) = 0, that g(n+1) =
0?

Does it then follow that the limit of (1/2, 3/4, 7/8, ....,
(2^n-1)/2^n, ...) is less than 1, because the limit of (g(0), g(1),
..., g(n)) is 0?

If it is, then the limit of the staircases is the line.

If it isn't, then how can the equation "y = mx + b" be a line? It's
just a specification that tells us a set of points satisfying an
equation.

Can you derive the formula a particular sequence (p_1, p_2, ...) which
has limit 0, just from knowing that the limit is 0?

And how can you talk about "differentiating", which is based on limits,
before we can actually agree on what a limit is? Differentiating is
/taking a limit/, and /not/ "taking the infinite case"; so if we can't
agree on what the limit of the staircases are, how can you be sure you
really know what "differentiating " is?

And n is presumably a number here - I think we can safely agree,
without going into details, that claiming that there is an "infinite" n
is, to say the least, controversial (and I might add, vague, in that I
have never seen a really complete description of what you intend this
to mean).

And finally - what has this to do with limits? In what way does your
statement illuminate the claim "the limit of C is X"?

R_k = ((x,y) : x = j/k, j in N and 0< j <= k; x - 1/k < = y <= x)

T_k = ((x,y) : y = j/k, j in N and 0 < j <= k; y - 1/k <= x <= y)

C_k = R_k union T_k.

To repeat the earlier argument:

Let p be any point on the diagonal. Let p_k be the instersection of the
line passing through p with slope 1 and the set C_k.

It is tedious to prove, but it should be fairly obvious that the
sequence (p_1, p_2, p_3, ...) is well-defined (i.e., there is a single,
unique point p_k for all k in N); and that furthermore this sequence
has limit p using the usual delta-epsilon proof; since the (Euclidean)
distance between p and p_k is always <= sqrt(2)/(2*k).

Therefore, every point p on the diagonal is a limit point of the
sequence C = (C_1, C_2, ..., C_n, ...); so D is at least a subset of
the limit of C.

Conversely, suppose p is not on the diagonal; then let d be the
distance from p to the diagonal. Then there is some n such that for all
m > n, the distance from the closest point on C_m to p is greater than
0; so it cannot be the case that there is a sequence of points {p_k}
where p_k in C_k such that the sequence has limit p; therefore p is not
a limit point of C if it is not a point in D.

Therefore every point of D is a limit point of C; and no point not in D
can be a limit point of C; so therefore the limit of C is exactly D.

Does this give us something to discuss?

A fractal is just a set of points in R^2, just like a line is; but of
course "fractal" isn't what you /really/ mean; it just sounds exotic
and strange; you mean something that has the properties that you
/think/ a fractal has; just as "the infinite case" is something that
has the properties that you /think/ a limit has.

I'm claiming that just because the diagonal has length sqrt(2), that
doesn't imply that therefore it is not the limit of the staircases.
Why? Because I know what the definition of "limit" is, and by that
definition, the staircase has limit the diagonal. And I know what the
length of the staircase is, and I know what thelength of the diagonal
is: these things are all just a matter of applying the definitions.

Where, in the definition of limit that I gave, do you see a reference
to "derivatives" or "deriving one formula from the other"?

For that matter, when you agree that the limit of {1/2^n} is 0, what
"derivative" are you taking?

When you agree that the limit of (a_1,a_2, ..., a_n, ...) is pi, how do
you "derive" the values of a_1, a_2, etc. from pi?