Prove that if f : (a,b)-->R is is uniformly continuous function in (a,b) so f is bounded function.
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?.
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.