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.