Function sequences in $C[0,1]$ using infinity norm

Question

I am working on the space of continuious function from $[0,1]$ to $\mathbb R$ with the infinity norm ($ \sup_{x\in [0,1]}|f(x)|$). My question is the following

Is it possible to construct a sequences of continuous function $f_{n}: [0,1] \rightarrow \mathbb R$ such that $\mid f_{n}(x)\mid$ converges to $0$ but the sequence $\{f_{n}\}_{n}$ does not converges to 0 in the normed space $C^{\infty}[0,1]$ with the infinity norm.

Answer 1

Hint: consider $f_n(x) = \begin{cases} nx &\text{ if } x\le \frac 1n\\ 2 - nx &\text{ if } \frac1n x\le \frac 2n\\ 0&\text{ otherwise} \end{cases}$

Prove that $f_n(x>0)$ converges to 0, and as $f_n\left(\frac 1n\right) = 1$.