Let X be a topological space and let Y be a metric space. Let fn: X...

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

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Answer:
We consider ε>0 and d = the metric of Y.
First of all, according to the uniform limit theorem => function f is continuous.
Secondly we recall the following theorem:
“If f:X -> Y is continuous then for every convergent...

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