Solution
= Solution
\b[False.] Take the uniformly continuous function $f(x)=x$. Then
$$
g_n(x)-f(x)^2=(x+1/n)^2-x^2=\frac{2x}{n}+\frac1{n^2},
$$
which is unbounded as a function of $x$ for every fixed $n$. Thus $g_n$ does not converge uniformly to $f^2$ on $\mathbb R$.