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Suppose that $X$ is a compact metric space. Let:

(a) $(f_n)$ be a sequence of real-valued continuous functions on $X$

(b) $(f_n)$ converges pointwise to a continuous function $f$ on $X$

(c) $f_n(x) \geq f_{n + 1}(x)$ for all $x \in X$ and all $n \geq 1$.

Show that $f_n \to f$.

How should I go about proving this?

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What have you tried so far? Where are you stuck? –  Andrew Uzzell Feb 8 '13 at 11:26
What do you mean by $f_n \to f$? Surely not pointwise convergence as this is a hypothesis. So maybe uniform convergence? –  user38355 Feb 8 '13 at 11:43

1 Answer 1

Hint: See Dini's Theorem. On Topics In Real Analysis, Theorem 3, p 193. Elementary Real Analysis, P 385. or Wikipedia.

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