Show that $f_n \rightarrow 0$ in $C([0, 1], \mathbb{R})$

I was given the following problem and was wondering if I was on the right track.

Let $ f_n(x) = \frac{1}{n} \frac{nx}{1 + nx}, \: 0 \le x \le 1$

Show that $ f_n \rightarrow 0$ in $ C([0, 1], \mathbb{R})$ .

I have this theorem that I figured I could use:

$ f_k \rightarrow f$ uniformly on A $ \iff$ $ f_k \rightarrow f$ in $ C_b$ .

In this case, $ C_b$ is the collection of all continuous functions on $ [0,1]$ . So if I can prove the function is uniformly continuous, this would prove that $ f_n \rightarrow 0$ . Can I apply this theorem like this to prove what I want? Also, if I can, would using the Weierstrass M test be best to prove uniform convergence here?