Solution
= Solution
This may be false when the limiting functions are unbounded. On $X=\mathbb R$, let
$$
f_n(x)=f(x)=x,
\qquad
g_n(x)=x+\frac1n,
\qquad
g(x)=x.
$$
Both sequences converge uniformly to their stated limits, but
$$
f_ng_n-fg=\frac xn
$$
is unbounded for every $n$. Thus the products do not converge uniformly.