Proving uniform convergence of $g(f_n(x))$

Proving uniform convergence of $g(f_n(x))$

Let $f(x),f_n(x):[a,b]\to[c,d]$ s.t $f_n\to f$ uniformly on [a,b]. Let
$g:[c,d]\to\mathbb R$ continious function. Prove that $g(f_n(x))\to
g(f(x))$
Let $\epsilon>0$. If $f_n\to f$, $\exists n_0\in\mathbb N$ such that for
all $n>n_0$ $|f_n(x)-f(x)|<\epsilon$. Since g is continious on [c,d],
exists $\delta>0$ s.t if $|x-x_0|<\delta$ then $|g(x)-g(x_0)|<\epsilon$.
In fact $\forall x\in [a,b]: f_n(x),f(x)\in[c,d]$ so if $n>n_0$ and
$|f_n(x)-f(x)|<\delta$ then $|g(f_n(x))-g(f(x))|<\epsilon$. I'm sure
something is really wrong with this proof (except it seems I proved it
only for specific environment of $x_0$). What is it and how can my proof
can be improved?