Let $f$ be a real uniformly continuous function on the bounded set $E$ in $\mathbb R$. Prove that $f$ is bounded on $E$.
What I tried to show was that by contradiction, saying that $f$ is unbounded on $ E$. Then we should be able to find an integer $N$ such that $f(N)>M$ for some integer $M$ which would bound the interval. I know by uniform continuity, that we can choose any epsilon we want. I get stuck about how to show this. Any hints would be wonderful. Thank you