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?


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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