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?.