Sequences of Functions - 國立臺灣大學
Sequences of Functions Uniform convergence 9.1 Assume that f n → f uniformly on S and that each f n is bounded on S. Prove that {f n} is uniformly bounded on S. Proof: Since f n → f uniformly on S, then given ε = 1, there exists a positive integer n 0 such that as n ≥ n 0, we have |f n (x)−f (x)| ≤ 1 for all x ∈ S. (*) Hence, f (x) is bounded on S by the following
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