## 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?

Thanks

## How do i compute $f_n = 3f_{n-1} + 2\sqrt{2f_{n-1}^2 – 2}$ for n around 10^18?

So I have the reccurence $$f_n = \begin{cases} 3f_{n-1} + 2\sqrt{2f_{n-1}^2 – 2}, &n > 0\ 3, &n > 1\ \end{cases}$$ and I need to compute it in $$\lg(n)$$, for n as big as $$10^{18}$$. I tried to reduce it to a closed form equation but I don’t see how that could be achieved.