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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?

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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 $k$.

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