What does "monotone functions are converging to a continuous limit" mean?

Question

In the book "Extreme Values, Regular Variation and Point Processes" by Resnick, it is used several times that "monotone functions are converging to a continuous limit". But what exactly does this mean? Considering the sequence $f_n(x)=x^n$ on the closed interval $[0,1]$ seems to be a counter example in case that the monotonicity refers to monotonicity with respect to $x$? I think this formulation caused also some confusion here; however, from there it still is not entirely clear to me what is meant. So in summary my question is, what the precise meaning is, in particular, is monotonicity meant with respect to $n$? And if yes it seems like we need at least continuity of each $f_n$? Also a reference for this statement would be nice. Thanks in advance!

Answer 1

The theorem requires monotonicity in $x$ not $n$. Continuity of the pointwise limit is an assumption rather than a consequence. Continuity of the $f_n$ is not required.

In full, let $f_n: [a, b] \to \mathbb {R}$ be a sequence of functions such that for all $x \in \mathbb {R}$ the limit $\lim_{n \to \infty} f_n(x)$ exists. Set $f(x) = \lim_{n \to \infty} f_n(x)$. Suppose that every $f_n$ is monotone and that $f$ is continuous. Then the convergence of $f_n$ to $f$ is uniform in $x$, i.e. $\sup_{x \in [a, b]} \lvert f_n(x) - f(x) \rvert \to 0$ as $n \to \infty$.

A proof of this is summarised well in the first answer of the question you have linked.