Consider
a mapping $a$ on $\mathbb{N} \mapsto (0,\infty)$ with $n\mapsto a(n)\equiv a_n$.
a mapping $b$ on $\mathbb{N} \mapsto \mathbb{R}$ with $n\mapsto b(n)\equiv b_n$.
Suppose that $\forall t\in \mathbb{R}$ and $\forall x \in \mathbb{R}$ $$ \lim_{n\rightarrow \infty} n \Big(1-G\Big(a_n(t+x)+b_n\Big)\Big)= \exp(-t-x) $$ where $G:\mathbb{R}\rightarrow [0,1]$ is continuous and strictly monotone increasing on $\mathbb{R}$.
Show that convergence is locally uniform with respect to $x$ $\forall t \in \mathbb{R}$.
As hint from the professor I was suggested to use result 0.1 in Resnick's book (Extreme Values, regular Variation, and Point processes) but I have some doubts.
First of all, let $$ U_n(x,t)\equiv n \Big(1-G\Big(a_n(t+x)+b_n\Big)\Big) $$ and $$ U_0(x,t)\equiv \exp(-t-x) $$ I think that local uniformity here intended as $$ \lim_{n\rightarrow \infty} \sup_{x\in [a,b]}|U_n(x,t)-U_0(x,t)|=0 $$ $\forall a\in \mathbb{R}, \forall b\in \mathbb{R}, a<b, \forall t \in \mathbb{R}$. Correct?
Secondly, result 0.1 n Resnick's book gives some sufficient conditions for local uniform convergence: for any $t\in \mathbb{R}$
$U_n(\cdot ,t)$ with domain $\mathbb{R}$
$U_n(\cdot,t)$ non decreasing on $\mathbb{R}$
$U_0(\cdot ,t)$ continuous on $\mathbb{R}$
$\lim_{n\rightarrow \infty} U_n(x,t)=U_0(x,t)$ $\forall x \in \mathbb{R}$
My doubt is related to the "non-decreasing part": my function $U_n(\cdot, t)$ is actually strictly decreasing on $\mathbb{R}$ because $G$ is strictly increasing in $x$ and $1-G(\cdot)$ is thus strictly decreasing in $x$. Can we extend Resnick's result to strictly decreasing functions? This and this questions are related (but the answers are given for non-decreasing functions!)