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Math Help - metric spaces

  1. #1
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    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??? ,,,
    Last edited by jin_nzzang; May 2nd 2009 at 04:55 AM.
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  2. #2
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    Quote Originally Posted by jin_nzzang View Post
    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 (x_n) be a sequence in X that converges to x. Then (x_n) is a Cauchy sequence in X. Use the given ε-δ condition to conclude that (f(x_n)) is a Cauchy sequence in Y. The completeness of Y tells you that (f(x_n)) 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 x_n\to x with f(x_n)\to w, as above, and also that y_n\to x with f(y_n)\to v. Then d(x_n,y_n)\to0, and therefore p(f(x_n),f(y_n))\to0, from which it follows that p(w,v)=0 and hence v=w.

    Thus for every sequence that converges to x, its image under f converges to w. Therefore \textstyle\lim_{y\to x}f(y) exists and is equal to w.
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