I feel like this is fairly obvious, but I can\'t seem to proveit. Let X and Y be

Question

I feel like this is fairly obvious, but I can't seem to proveit.

Let X and Y be metric spaces and f: X ---> Y.
Assume that whenever the sequence {xi} converges tox0 then {f(xi)} converges.

Prove that if the sequence {xi} converges toxo (in X) then the sequence {f(xi)} convergesto f(xo).

I tried a proof by contradiction, but I keep getting stuck. Will rate.

Explanation / Answer

This result is true only if the function f is continuous. So we assume f is a continuous function. We have that xi ---> x. Now we provef(xi) converges to f(x). (That is for every > 0, we have to find an integer N suchthat |f(xi)-f(x)| < . ) Let > 0 be given.      Since f is continuous at x for there exists a >0 such that    |f(x)-f(y)| N we have that |xi -x|

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