Prove $F_n$ converges uniformly.

Question

Let $(f_n)$ be a sequence of continous functions which converges uniformly on $[a, b]$ to a function $f$.

Define:

$F_n(x) = \frac{f_1(x) + f_2(x)+...+f_n(x)}{n}$.

Prove:

$F_n$ converges uniformly to $f$ on $[a,b]$.

Im not so sure how to approach the question.

Any attempts to use Weierstrass M-test failed.

How can i continue?

Hints will be appericiated.