Let $(f_n(x))_{n=1}^{\infty}$ a series of uniformly continuous functions, $\mathbb{R}\to\mathbb{R}$ which uniformly converges to the function $f$, and a continuous function $g$ :$\mathbb{R}\to\mathbb{R}$.
I need to give an example where $(g\circ f_n)_{n=1}^{\infty}$ is not uniformly converges and to prove that if $g$ would be uniformly continuous so $(g\circ f_n)_{n=1}^{\infty}$ would uniformly converges.
As an example I gave $g=\sin x$ and $f_n= \frac{1} {n+x^2}$, but I'm having a hard time proving the claim.
Any hints? Thanks!