# Dini's theorem: changing the monotonicity requirement

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.