The following is a proof from "Heavy-Tail Phenomena" by Resnick (2007). I have some questions about the proof.
- (2.3) seems to be an identity. The left side the global sup over $[a, b]$ and hence should be greater than or equal to the right side which is the maximum of local sups. Right?
- I do not get the last set of inequality. Do we have that $U_0$ is also non-decreasing since $U_n \to U_0$ and $U_n$ in non-decreasing? And how to use (2.2) exactly there? Thank you!
- By the way, is it OK to post the picture here? Any problem with copyright?
Take an arbitrary $\varepsilon>0$.
$[a,b]$ is compact, so $U_0$ is uniformly continuous. Therefore you can take some points $a=x_0<x_1<\ldots<x_k=b$ such that $\max U_0-\min U_0<\frac\varepsilon2$ in every interval $[x_{i-1},x_i]$.
There is an index $n_0(\varepsilon)$ such that for $n>n_0$ and all $i$, $|U_n(x_i)-U_0(x_i)|<\frac\varepsilon2$. Then for exery $i$ and all $x\in[x_{i-1},x_i|]$, we have $$ U_n(x) \ge U_n(x_{i-1}) > U_0(x_{i-1})-\frac\varepsilon2 > U_0(x) - \varepsilon $$ and $$ U_n(x) \le U_n(x_i) < U_0(x_i)+\frac\varepsilon2 < U_0(x) + \varepsilon. $$
Hence, for arbirary $\varepsilon>0$ there is an index $n_0$ such that for every $n>n_0$ and for all $x\in[a,b]$ it holds that $|U_n(x)-U_0(x)|<\varepsilon$.