A sequence converges uniformly

Question

If a sequence $\{f_n\}$ converges to $f$ uniformly in $\mathbb{R}$, does it follow that $(f_n)^2$ converges to $f^2$ uniformly?

Answer 1

Edit : This is true if $f$ is bounded, as pointed out by @PedroTamaroff.

Yes, since $(f_n)$ is uniformly bounded (why?) and $$ |f_n^2(x) - f^2(x)| = |f_n(x) -f(x)||f_n(x) + f(x)| $$

Answer 2

This is not true. Consider $f_n(x)=x+n^{-1}$