If is not uniformly continous then for there is such that for any we have , but .
Let for .
Then we form two sequences and such that and . Since is a bounded sequence by Bolzano-Weierstrass theorem there is a convergent subsequence . Let and note that . Let be the subsequence chosen in the same manner as i.e. if for some choice function then . Now since we see that . But each converges to and so by squeeze-theorem converges to . Thus, and are two convergent sequences to . By continuity it means and converge both to . However, and this is a contradiction.