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

Question

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?

Thanks

Answer 1

You'll notice that you can cancel to get $f_n(x) = \frac{x}{1+nx}$. For $x =0$, clearly $f_n(x)= 0$. Otherwise, $nx$ gets arbitrarily large. This will imply that $f_n(x) \to 0$.

Answer 2

note that for $ x \in [0,1] $ we have $|f_n(x)| =|\frac{1}{n}\frac{x}{1+nx}| \le \frac{1}{n}$, so $||f_n||_\infty \le \frac{1}{n}$ which implies $f_n \to 0 $ in $C_b[0,1]$