TITLE: Sufficient condition under which a pointwise convergent becomes uniform convergence
QUESTION [6 upvotes]: Let $K$ be any compact subset of $\mathbb{R}$. Then what are the
  sufficient conditions so that any pointwise convergent sequence of functions on $K$ converges uniformly.

The conditions can be given as in Dini's Theorem. Can we have other conditions (Weaker) apart from Dini's theorem so that this becomes true?

REPLY [2 votes]: Another standard theorem is the following:
Let $f_k \colon [a,b]\to\mathbb{R}$ be a sequence of functions, such that $f_k$ is  non-increasing (resp. non-decreasing) for every $k\in\mathbb{N}$.
If $(f_k)$ converges pointwise to a continuous function $f\colon [a,b]\to\mathbb{R}$, then $f$ is non-increasing (resp. non-decreasing) and the convergence is uniform.
Note that in this theorem the monotonicity is in the $x\in [a,b]$ variable, whereas in Dini's theorem is on the index $k$.