Let be a sequence in X that converges to x. Then is a Cauchy sequence in X. Use the given ε-δ condition to conclude that is a Cauchy sequence in Y. The completeness of Y tells you that converges to some element w in Y.

It remains to show that for every sequence that converges to X, its image under f converges to the same limit in Y. So suppose that with , as above, and also that with . Then , and therefore , from which it follows that and hence v=w.

Thus for every sequence that converges to x, its image under f converges to w. Therefore exists and is equal to w.