Thread: metric spaces

Let (X,d) and (Y,p) be metric spaces, and assume Y is complete. Let f: X→Y and x∈X.
Show that f has a limit at x if and only if given any ε>0 there exists δ>0 such that whenever y, z ∈B_δ(x)＼｛x｝,p(f(y),f(z)) < ε.



can anyone please help me to solve this question??? ,,,

Originally Posted by jin_nzzang

Let (X,d) and (Y,p) be metric spaces, and assume Y is complete. Let f: X→Y and x∈X.
Show that f has a limit at x if and only if given any ε>0 there exists δ>0 such that whenever y, z ∈B_δ(x)＼｛x｝,p(f(y),f(z)) < ε.

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.