Re: Cantor Confusion

Andy Smith wrote:
I don't know about the cos(m! pi x) term, but (bearing in mind I don't
have a pilot's licence) I would say that:

Lim n->oo {|x|^n} is definitely not 0 for |x|<1, 1 for |x|=1.

At a first glance one is tempted to say, well, at any n, let x >=
1-1/(2n). Then for any n, x^n >1/2 (terms of (1-y)^n are monotonic
decreasing for y<1 and alternating in sign), so
it cannot be true that the lim n->oo of |x|^n = 0 for all |x| <=x<1

You have demonstrated what people on sci.math refer to as dyslexia. You
have switched the order of the operations. As far as the limit is
concerned, x is a fixed number. Let's try x = 1/2. The question is what
is the value of

lim n->oo (1/2)^n

? To be more precise, saying

lim n->oo |x|^n = 0, for |x| < 1

is different from saying

lim n->oo sup_{|x|<1} |x|^n.

"sup" is like max, but is used when the maximum isn't obtained.

--
David Marcus
.

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