I would appreciate any help that I can get with the following proof:

Let $\displaystyle f_n,f:[0,1]\rightarrow \mathbb{R}$ and $\displaystyle x,x_n\in [0,1]$ such that each $\displaystyle f_n$ is continuous. Also, $\displaystyle f_n\rightrightarrows f$ and $\displaystyle x_n\rightarrow x$.

Prove that $\displaystyle f_n(x_n)\rightarrow f(x)$.

So i think my goal is to find an $\displaystyle N$ such that $\displaystyle n\geq N\Rightarrow |f_n(x_n)-f(x)|<\epsilon$.

Also, we know that there exists an $\displaystyle N$ such that $\displaystyle n\geq N\Rightarrow |f_n-f|<\epsilon$ and also, $\displaystyle |x_n-x|<\epsilon$.

Assuming I'm correct so far, I need some help putting it all together...