We need to get down a few facts about metric spaces before we can continue on our course. Firstly, as I alluded in an earlier comment, compact metric spaces are sequentially compact — every sequence has a convergent subsequence.
To see this fact, we’ll use the fact that compact spaces are the next best thing to finite. Specifically, in a finite set any infinite sequence would have to hit one point infinitely often. Here instead, we’ll have an accumulation point in our compact metric space so that for any and point in our sequence there is some with . That is, though the sequence may move away from , it always comes back within of it again. Once we have an accumulation point , we can find a subsequence converging to just as we found a subnet converging to any accumulation point of a net.
Let’s take our sequence and define — the closure of the sequence from onwards. Then these closed sets are nested , and the intersection of any finite number of them is the smallest one, which is clearly nonempty since it contains a tail of the sequence. Then by the compactness of we see that the intersection of all the is again nonempty. Since the points in this intersection are in the closure of any tail of the sequence, they must be accumulation points.
Okay, that doesn’t quite work. See the comments for more details. Michael asks where I use the fact that we’re in a metric space, which was very astute. It turns out on reflection that I did use it, but it was hidden.
We can still say we’re looking for an accumulation point first and foremost, because if the sequence has an accumulation point there must be some subsequence converging to that point. Why not a subnet in general? Because metric spaces must be normal Hausdorff (using metric neighborhoods to separate
closed sets) and first-countable! And as long as we’re first-countable (or, weaker, “sequential”) we can find a sequence converging to any limit point of a net.
What I didn’t say before is that once we find an accumulation point there will be a subsequence converging to that point. My counterexample is compact, and any sequence in it has accumulation points, but we will only be able to find subnets of our sequence converging to them, not subsequences. Unless we add something to assure that our space is sequential, and metric spaces do that.
We should note in passing that the special case where is a compact subspace of is referred to as the Bolzano-Weierstrass Theorem.
Next is the Heine-Cantor theorem, which says that any continuous function from a compact metric space to any metric space is uniformly continuous. In particular, we can use the interval as our compact metric space and the real numbers as our metric space to see that any continuous function on a closed interval is uniformly continuous.
So let’s assume that is continuous but not uniformly continuous. Then there is some so that for any there are points and in with but . In particular, we can pick as our and get two sequences and with but . By the above theorem we can find subsequences converging to and converging to .
Now , which converges to , and so . Therefore we must have also converging to by the continuity of . But this can’t happen, since each of these distances must be at least ! Thus must have been uniformly continuous to begin with.