Let $\displaystyle (f_n)^\infty_{n=1}$ be a sequence of functions which are defined on $\displaystyle [a,b]$, and uniformly converges there to bounded function $\displaystyle f$. Let $\displaystyle \Phi\in C(\mathbb{R})$ which is continuous function on all $\displaystyle \mathbb{R}$.

Prove that the sequence $\displaystyle (\Phi(f_n))^\infty_{n=1}$ uniformly converges and find the limit function.

Thank you!