I've just studied Dini's theorem, and I've been thinking.

Dini's Theorem:

Let $f_n:[a,b]\rightarrow \mathbb{R}$ be a sequence of continuous functions such that $f_n\rightarrow f$ pointwise.

Suppose $f_n(x)$ is a decreasing sequence for all $x$ and $f$ is continuous.

Then $f_n \xrightarrow{u} f$

The monotonicity requirement promises that the "peak" of the sequence of functions won't "run" to inifinity as $n\rightarrow \infty$.

I'm trying to understand what this requirement can be replaced with.

My intuition tells me that it should be something like uniform continuity, but stronger than that.

Would appreciate if you could share your thought about this matter.

  • $\begingroup$ What version of Dini's theorem do you mean? Is it the one from Wikipedia : en.wikipedia.org/wiki/Dini%27s_theorem ? $\endgroup$ – Olivier Bégassat Apr 24 '17 at 9:19
  • $\begingroup$ You are probably looking for something like equi-continuity (which avoids monotonicity). Take a look are Arzela -Ascoli Theorem (found almost everywhere on the net and in books :-)). $\endgroup$ – Kavi Rama Murthy May 9 '18 at 6:05

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