Let $A$ be any family of monotonic increasing functions defined on the closed interval $[a,b]$. There exists an $M$ such that for any $x$ in $[a,b]$ and any $f$ in $A$, we have:
$|f(x)| \leq M$

Prove:
There exists a sequence of functions $f\_n(x)$ in A and a monotonic increasing function $g(x)$ such that $f\_n(x) \to g(x)$ if $n \to \infty$ for any $x$ in $[a,b]$.