Let be a continuous function such that for each it holds
Prove that is uniformly continuous.
Firstly, let us define the following set:
By hypothesis, for any fixed , we have
Note that each set is closed because for a fixed , the set of values of such that is closed and is an intersection of these closed sets over all .
Note that we could also say that
since the sets increase with – giving a countable union of closed sets whose union is the whole space.
We can then apply the Baire Category Theorem to say that since a countable union of closed sets has non-empty interior, some element of the union must have an interior! In particular, for any , there must be some such that some interval is a subset of . However, then if we have that and we could choose some pair with and then observe that
Then the distance from to is at most as is the distance from to since . Thus we find that if and we have – and this works out for any choice of . This fact suffices to establish that is uniformly continuous with a small bit of further work.