I know it has been asked many times to show that the space $\mathcal{C}^0([0,1])$ is not complete with respect to the $L^1$-norm, but I was given to show it using one particular sequence and at the moment I think the problem statement is false.
I was asked to consider the sequence $(f_n)_{n \in \mathbb{N}}$ with
$$f_n: [0,1] \rightarrow \mathbb{R} \hspace{30pt} x \mapsto \begin{cases} xn & 0 \leq x < \frac{1}{n} \\ 2 - xn & \frac{1}{n} \leq x < \frac{2}{n} \\ 0 & \frac{2}{n} \leq x < 1\end{cases}$$
to show that the given space is not complete, but I feel confident that the sequence of functions will converge to the constant zero-map in the $L^1$-Norm. Therefore this sequence is not suitable for the proof.
Am I missing something?