(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

For some reason, although it looks simple, it's giving me trouble.

Let X be a topological space, and Y a metric space. Let fn : X --> Y be a sequence of continuous functions, and let xn be a sequence of points in X converging to x. Show that if fn converges uniformly to f, then (fn(xn)) converges to f(x).

3. The attempt at a solution

The facts I know:

i) since fn is a sequence of continuous finctions which converges uniformly to f, f is continuous

ii) since f is continuous, f(xn) converges to f(x).

Now, since fn converges uniformly to f, for every ε > 0 there exists some N such that for all x in X and for any n >= N, d(fn(x), f(x)) < ε holds. (iii)

I need to show that for any ε > 0, there exists some integer N such that, for all n >=N d(fn(xn), f(x)) < ε holds.

Let ε > o be given. Since, because of uniform convergence (iii) holds for any x, it holds for the members of the sequence xn, too. So, there exists N such that for n >= N, d(fn(xn), f(xn)) < ε holds. Now I'm stuck. Could this mean that the sequences fn(xn) and f(xn) converge to the same limit? Since then fn(xn) would definitely converge to f(x). But I can't find a theorem or result which says anything about that right now.

Perhaps I'm not on the right track at all. Thanks for any replies.

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# Another sequence problem

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