Re: Cantor Confusion

David Marcus writes

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.


But, if the formula is to be true for all x<1, then don't we need to show that

lim n->oo sup_{|x|<1} |x|^n = 0 ?

Which was what I was trying, unsuccessfully and amateurishly, to do/disprove? that is exactly the point at issue - there is no question that
lim n->oo |x|^n = 0 for any |x| that is finitely different from 1.

The issue is whether there are any wrinkles and whether the limit is true for
all |x| strictly <1 i.e. all reals in the interval from 0,1 excluding the exact point at x = 1.



--
Andy Smith
.

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