Prove that if f : (a,b)-->R is isuniformly continuous functionin (a,b) so f is bounded function.

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- Jan 3rd 2010, 10:11 AMAlso sprach ZarathustraUniformly continuous #3
Prove that if f : (a,b)-->R is is

**uniformly continuous function**in (a,b) so f is bounded function. - Jan 3rd 2010, 01:28 PMJose27
Assume it's not bounded then pick a (unbounded) sequence such that for all then since this sequence is bounded we get a Cauchy subsequence by Bolzano-Weierstrass. Since is uniformly cont. it sends Cauchy seq. into Cauchy seq. but what can you say about by construction?.

- Jan 3rd 2010, 01:56 PMPlato
- Jan 3rd 2010, 01:58 PMAlso sprach Zarathustra
hmmm... I don't understand...

What is Cauchy subsequence ?

"Since f is uniformly cont. it sends Cauchy seq..." why is that? - Jan 3rd 2010, 02:55 PMAbu-Khalil
- Jan 3rd 2010, 02:56 PMJose27
- Jan 3rd 2010, 03:29 PMAlso sprach Zarathustra
Dear Jose! Can you please write me why the existence of such K ==> boundedness of f?

- Jan 3rd 2010, 03:43 PMJose27
So you know a Cauchy sequence is bounded, and

**Abu-Khalil**proved that if is a unif. cont. function and is a Cauchy seq. then is also a Cauchy seq. but by how we constructed them (or in which case invert all inequalities in my first post) clearly contradicting the fact that is unif. cont. so we conclude that our initial hypothesis is false ie. is bounded.