Clearly for all t>0 (because in fact ). So we need to show that given t>0 there exists n such that . This seems to be quite tricky to prove. One way that works (although it's not very elegant) is to suppose that the result is false, and get a contradiction.

So suppose that there exists t>0 such that for all n there is an element with . Use compactness to conclude that there is a subsequence of that converges, to say. Then get a contradiction by showing that but also .