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Definition: Let f_n: X -> Y be a sequence of functions from the set X to the metric space Y. Let d be the metric for Y. We say that the sequence (f_n) converges uniformly to the function f: X-> Y if given e (epsilon) greater than 0, there exists an integer N such that d(f_n(x), f(x)) < e for all n > N and all x in X.
Source: Topology (second edition) by James R. Munkres