Here's my attempt:

Assume to the contrary that is a uniformly continuous function on a bounded set but is unbounded on . Then there exists a sequence such that for all . Since is a bounded sequence, there exists a subsequence that is a Cauchy sequence. Since is uniformly continuous and is a Cauchy sequence in , then is a Cauchy sequence. So converges to some real number. But for all as well, which is a contradiction. Q.E.D.

Is this correct? Thanks.