Let be a continuous function and . Show that is uniformly continuous.

I'd like to use this charactetization of uniform continuity. A function is uniformly continuous iff for all sequences if , then also .

Suppose .

If is a bounded sequence, then must be bounded too, and so if , then also , because is continuous and therefore uniformly continuous on a compact set. If, on the other hand, , then also and we have

But what if is neither bounded nor convergent to infinity?