Re: Analysis problem, need some help here...

From: José Carlos Santos (jcsantos_at_fc.up.pt)
Date: 11/08/04 

Date: Mon, 08 Nov 2004 18:24:40 +0000

damsel-in-distress wrote:

> So you are using the proof of contradiction, but how does the fact
> that (a_n)_n has some convergent subsequence contradict anything,
> so as to make the initial hypothesis false? I can be a lil slow at
> times.. :)

First of all, two things:

1) Please don't top-post. If you want to know what's that and why you
shouldn't do it, read

http://www.caliburn.nl/topposting.html

or

http://www.html-faq.com/etiquette/?toppost

for instance.

2) While writing, please press the ENTER key once in a while.

Now, concerning your specific question: let (b_n)_n be a convergent
subsequence of (a_n)_n and let l = lim_n b_n. Then l belongs to [a,b]
and lim_n f(b_n) = inf f = 0. But, since f is continuous at l, you
have f(l) = f(lim_n b_n) = lim_n f(b_n) = 0, which is impossible.

Best regards,

Jose Carlos Santos