> 
> Given normed space $C[0;1]$ (the space of continous functions defined on $[0;1]$) with the norm $||.||$ defined as following
> 
> 
> $$||x|| = \max\limits\_{x \in [0;1]} |f(x)| $$
>  Prove that the sequence $\{x\_n\}\_n$ doesn't converge in $(C[0;1],||.||)$ with 
>  $$x\_n(t) = \left\{ \begin{array}{ll}
>  nt,& 0 \le t \le \dfrac{1}{n} \\ 
>  1,& \dfrac{1}{n} < t \le 1
> \end{array} \right. $$
> 
> 
> 

In the attempt of solving this problem, I found out that the sequence converges by individual points to the function $x$ (means for every $t \in [0;1]$, $\lim\limits\_{n \to \infty} x\_n(t) = x(t)$) with
$$x(t) = \left\{ \begin{array}{ll}
 0,& t = 0 \\ 
 1,& 0 < t \le 1
\end{array} \right. $$
which isn't an element of $C[0;1]$.

This might be a point in the solution. However, I don't know how to rigorously proceed with the correct definition of the coverage in a normed space (as $x\_n - x$ isn't an element of the normed space). Please help me. Thank you for reading.