Re: uniqueness of limits at infinity

From: David C. Ullrich (ullrich_at_math.okstate.edu)
Date: 09/21/04 

Date: Tue, 21 Sep 2004 05:46:25 -0500

On Mon, 20 Sep 2004 23:21:45 -0300, "Shmuel (Seymour J.) Metz"
<spamtrap@library.lspace.org.invalid> wrote:

>In <d3be1825.0409200915.130b1e3f@posting.google.com>, on 09/20/2004
> at 10:15 AM, jani@persian.com (Jani Yusef) said:
>
>>That is exactly what I meant. What particular theorem holds this to
>>be true?
>
>No theorem; limits in real analysis are unique by definition.

Uh, no. The definition of "lim_{x->infinity} f(x) = L" is that
for every eps > 0 there is N so that |f(x) - L| < eps for all
x > N. There's no reason a priori that this could not hold
for two different values of L; the fact that that's impossible
needs to be proved. (Yes, the proof is pretty easy. But it's
not just true by definition, it's a theorem in books. Honest.)

************************

David C. Ullrich

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