If $\{f_{n}(x)\}$ is a sequence of non-zero real valued functions, which are continuous and bounded by some constant $A>0$. Let $a_{n}=\sup_{x\in \mathbb R}|f_{n}(x)|$ and the sup is attained at the points $p_{n}$, also $\lim_{n\to\infty}a_{n}=0$. Define a new function $F_{n}(x)=\frac{f_{n}(x)}{a_{n}}$, for all $n\geq 1$, and $x\in \mathbb R$. What can we say about $\lim_{n\to\infty}F_{n}(x)$? Is it zero? If not when it could be zero?
My guess is we cannot know! Unless $f_{n}$ converges to $0$ faster that $a_{n}$, but the given information doesn't tell if this is the case or not!
Looking to this from another point of view: Note that $1=\sup|F_{n}|$, so if $F_{n}\to 0$, then we must have $\sup |F_{n}|\to 0$, a contradiction!!