Ok, so lets just establish a few things, the metric in is . Since assuming that we have that is a real function. So for to be uniformly continous we must have that for every there exists a such that .

1. Now assume the converse and state that is unbounded. By the fact that we are mapping to the reals let us fix and let be finite. Now since is unbounded in for every we may find a such that , so for every we can find a such that which violates