Uniform continuity

Heine-Cantor theorem

Let be a continuous function between metric spaces and be a compact. Then is uniformly continuous. anal

Proof

Let and be a metric spaces with compact, with a continuous mapping between them. Let . By continuity, for every there exists such that for all . Then the balls form a open cover of , so by compactness there must exist a finite set of points whose balls cover the space, i.e. is a finite subcover. Then there exists since it is the minimum of finitely many positive real numbers.

Now we will show that meets the requirements for Uniform continuity. Let such that . Then for some . Then by the triangle inequality

Therefore and thus by the original definition of it follows . Thus as required.

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