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Let $\{f_n\}$ be a uniformly bounded sequence of continuous functions such that $f_n(x) \leq f_{n+1}(x)$ for all $x \in X$ where $X$ is compact. Prove $\{f_n\}$ converges uniformly to some function $f: X \to \mathbb{R}$.

For starters, we can use monotonicity and the uniformly boundedness to show point-wise convergence. By Dini's theorem, the convergence is uniform if and only if $f$ is continuous. I'm having some trouble showing that the convergence is uniform or that $f$ is continuous.

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    $\begingroup$ What about $f_n(x) = x^{1/n}$ on the space $X = [0,1]$? $\endgroup$ – Mr. Chip Sep 21 '17 at 22:55
  • $\begingroup$ That really does seem like a counter example, doesn't it. I pulled the question from a comprehensive exam so I forgot to actually look for one. $\endgroup$ – GillyB Sep 21 '17 at 23:06
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    $\begingroup$ Yeah, the convergence definitely isn't uniform, and nor is your limit function continuous. $\endgroup$ – Mr. Chip Sep 21 '17 at 23:10

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