Dini's monotone convergence theorem says that if

  1. $X$ is compact
  2. $f_n:X\to\mathbb{R}$ is continuous for all $n\in\mathbb{N}$
  3. $f_n\to f$ pointwise, with $f$ continuous
  4. $f_n(x)\leq f_{n+1}(x)$ or $f_n(x)\geq f_{n+1}(x)$ for all $x\in X$

then $f_n\to f$ uniformly.

Can (4) be relaxed so that the monotonicity is allowed to change direction with $x$? That is, if we say a series of functions $\left\{f_n\right\}$ is pointwise monotone if the series $\left\{f_n(x)\right\}$ is monotone (either increasing or decreasing) for each $x\in X$, can we replace (4) with the statement that $\left\{f_n\right\}$ is pointwise monotone?

(If there is a standard term for what I have called "pointwise monotone," please tell me.)

  • $\begingroup$ Let $f_n: [0,1] \to \mathbb{R}, f_n(x):=x^n$ for each $x \neq 1$ with have $\{f_n(x)\} \downarrow 0$, clearly $\{f_n(1)\}$ is a constant sequence, and yet the functions obviously do not converge uniformly, as the limit function is not continuous. $\endgroup$ – mb- Aug 12 '16 at 1:36
  • $\begingroup$ The question assumes the limit is continuous in 3. $\endgroup$ – Thompson Aug 12 '16 at 1:37
  • $\begingroup$ My mistake, you're right! $\endgroup$ – mb- Aug 12 '16 at 1:38

This might be true... Maybe you can do it like this: Look at the compact set $\{x \in X : f(x) \geq f_1(x)\}$. Here you have to see $f_n$ increasing or else they wouldn't converge even pointwise. (Note if $f(x) = f_1(x)$ then $f_n(x) = f_1(x)$ for all $n$.) So you can apply usual Dini. Do the same for the closed set $\{x \in X : f(x) \leq f_1(x)\}$.

  • $\begingroup$ Ah, yes -- duh! Thanks. $\endgroup$ – symplectomorphic Aug 12 '16 at 2:31

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