Convergence of composition of sequence of functions on a compact subset of $\mathbb{R}$. Let $\lbrace f_n\rbrace$ and $\lbrace g_n\rbrace $ be two sequences of real-valued continuous functions on a compact subset of $\mathbb{R}$ such that they converge pointwise to $f$ and $g$ respectively, where $f$ and $g$ are continuous functions on $\mathbb{R}$. Then what can we say about the composite function $f_n \circ g_n$. Can we say that it will converge to $f\circ g$.

A similar question is asked here Convergence of composition of functions sequences. But here the continuity of functions $f$ and $g$ was not given. Please at least give me some hint to solve this.
Let $f_n(x)=nx$  for $0 \leq x \leq 1/n$, $2-nx$  for $ 1/n \leq x \leq 2/n$ and $0$ for $x &gt;2/n$. $g_n(x)=\frac 1 n$. Let $f(x)=0$ for all $x$ and $g(x)=0$ for all $x$. Then $f_n(g_n(x))\to 1\neq f(g(x))$.
Let $\lbrace f_n\rbrace$ and $\lbrace g_n\rbrace $ be two sequences of real-valued continuous functions on compact subset of $\mathbb{R}$ such that they converge pointwise to $f$ and $g$ respectively, where $f$ and $g$ are continuous functions on $\mathbb{R}$. Then what can we say about the composite function $f_n \circ g_n$. Can we say that it will converge to $f\circ g$?