Constructing sequence of functions I have to construct a sequence of $\{f_i\}$, where $f_i$ belongs to $C[0,1]$ such that:
$$
d(f_i,0) = 1 \\
d(f_i,f_j)=1, \forall i,j \\
\text{Using Sup-Norm metric, i.e.} \mathbb{\|}f\mathbb{\|} = \sup_x{\mathbb{|}f_i (x)\mathbb{|}}
$$
Thanks.

Set
$$
f_n(x)=\left\{
\begin{array}{lll}
0 & \text{if}& x\le \frac{1}{n+1}, \\
 n(n+1)x-n& \text{if} & x\left[\frac{1}{n+1},\frac{1}{n}\right], \\
1 & \text{if}& x\ge \frac{1}{n}. 
\end{array}
\right.
$$
Clearly,
$$
\sup_{x\in[0,1]}|f_m(x)-f_n(x)|=1,
$$
if $m\ne n$.
