Re: Cantor Confusion

David Marcus writes

I can see that I am irritating you.

No, you aren't.

It is not deliberate. It is a
consequence of a misunderstanding on my part of something fundamental.

I'm sure!

You say, is there such a y? Well, if we consider the mapping
w = Lim n->oo x^n, 0<=x<=1 , we can consider that as the limit of a
succession of mappings.

This is simply a language problem. The meaning of what is written is not
what you think. It is possible to write what you are thinking: see
below.

An analogy would be if I tell you what the word "cat" means and you ask
why can't a cat be the animal that barks. It could be, but it isn't. We
have another name for that animal.

So, for example we can think of some
representative points located in x

Perhaps I should have said in {x} 0<=x<=1 then.

You can't say "in x". x is a number. We don't know exactly which, but it
is still a number. So, just as you can't say "in 7", you can't say "in
x". "x" is a name for a number, the number x. Similarly, "7" is a name
for the number 7.

e.g x - 0,1/4,1/2,1 on [0,1]. After
x^2 we get
0,1/16,1/4,1 and after x^n these points map to 0,1/(4^n),(1/2^n), 1.
But we still have a continuum of points in [0,1] after n iterations, and
each of these map back to an original location in x. And that has to be
true after n->oo

There is no "after n -> oo".

- the line is infinitely elastic. Otherwise you could
define a space between two reals in x?

There is then an inverse mapping from the points in w in [0,1] back to
their original locations in x? And these would all describe the y that
you request?

Name a number y with |y| < 1 that will work. You can't.

The following is from Chapter 24 of the book "Calculus" by Spivak.

"Let {f_n} be a sequence of functions defined on the set A. Let f be a
function which is also defined on A. Then f is called the uniform limit
of {f_n} on A if for every e > 0 there is some [natural number] N such
that for all x in A,

if n > N, then |f(x) - f_n(x)| < e.

"We also say that {f_n} converges uniformly to f on A, or that f_n
approaches f uniformly on A.

"As a contrast to this definition, if we only know that

f(x) = lim_{n->oo} f_n(x) for each x in A,

then we say that {f_n} converges pointwise to f on A. Clearly, uniform
convergence implies pointwise convergence (but not conversely!)."

Another way to write that {f_n} converges uniformly to f on A is

lim_{n->oo} sup_{x in A} |f_n(x) - f(x)| = 0.

Let f_n(x) = x^n. Let f(x) = 0. Then {f_n} converges pointwise to f on
[0,1), but not uniformly.

Thanks for taking the time. Spivak should be here soon ...

My mental problem with the limit of x^n was that viewed as mapping of points I couldn't see how you could split [0,1] into [0,1) and 1 by any sort of limit process - I had a mental image of x^n as a y-axis and x as the x-axis. At n = 1 it is a line, then progressively curving close to the x-axis as n increases, but always continuous. Viewed from the y-axis, there always seemed to be an inverse mapping.

But I can see now that this is far too simplistic, and I accept the proof. Ta.

--
Andy Smith
.

Relevant Pages

  • Re: Cantor Confusion
    ... that doesn't matter - we have plenty of reals. ... There is then an inverse mapping from the points in w in back to ... function which is also defined on A. Then f is called the uniform limit ... convergence implies pointwise convergence." ...
    (sci.math)
  • Mapping in complex plane
    ... is a conformal mapping that "looks" a bit like one I am actually ... I would like to have the "tall" rectangle one unit wide map onto ... a shape that approaches a quarter circle with unit radius. ... points on the x-axis and y-axis should remain on those axes. ...
    (sci.math)
  • Mapping in complex plane
    ... is a conformal mapping that "looks" a bit like one I am actually ... I would like to have the "tall" rectangle one unit wide map onto ... a shape that approaches a quarter circle with unit radius. ... points on the x-axis and y-axis should remain on those axes. ...
    (sci.math.symbolic)