Let and be metric spaces, and let . Prove that f is an uniformly continuous mapping on A iff for every pair of sequence of A with , we have

Proof.

Suppose that f is uniformly continuous, pick sequences of A such that

Let be given, and let , then such that whenever , we have . Since f is uniformly continuous, we then ahve , implies that

Conversely, suppose that math] \{ x_k \} , \{ y_k \} [/tex] are sequences in A with , we have .

Let be given, pick , find such that whenever , we have .

Now suppose that and , then , but we also have , which means .

Q.E.D.

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