$\inf$ of a sequence of $\sup$

Question

let $f_n$ a sequence of right-continuous functions $[0,\infty) \to R$. Assume that for each $T$: \begin{align} & \sup_{\substack{t \in [0,T]\\n \in N}} |f_n(t)|<\infty \end{align}

Is it true that for each $T$: \begin{align} & \inf_{a>0} \sup_{\substack{t \in [0,T+a]\\n \in N}} |f_n(t)|=\sup_{\substack{t \in [0,T]\\n \in N}} |f_n(t)| \end{align}

My problem is that the $f_n$ need not be equi right-continuous.

Answer 1

Let $f_n$ be continuous, equal $0$ on $[0,1]$, equal to $1$ on $[1+1/n,\infty)$ and affin-linear on $[1,1+1/n]$. For $T=1$ and every $a>0$ you then have $$\sup_{t\in [0,1+a],n\in\mathbb N} |f_n(t)| =1 \text{ and } \sup_{t\in [0,1],n\in\mathbb N} |f_n(t)| =0.$$