Why is $(C([a,b]),||\cdot||_{1})$ not complete?

Question

Question: I'm supposed to show that $(C([a,b]),||\cdot||_{1})$ with $||\cdot||_{1}:=\int^{b}_{a}|f|dx$ is not complete.

My attempts: I am trying to find a function sequence $f_{n}:[0,1]\rightarrow \mathbb{R}$ in $(C([0,1]),||\cdot||_{1})$ that is a Cauchy sequence but that diverges regarding $||\cdot||_{1}$. Is this correct?

I can think of countless examples that either converge with respect to $||\cdot||_{1}$ OR are Cauchy but I can't find any sequences that are BOTH. One strategy I've been trying is to push a function into the domain that isn't defined on the whole domain e.g. $f_{n}:[0,1]\rightarrow\frac{1}{x-\frac{1}{n}}$ but until now still in vain.

Any ideas would be greatly appreciated :)

Answer 1

HINT:

You don't need a sequence $(f_n)$ that diverges altogether. It will be enough to find a sequence that converges in $||\cdot||_1$ but whose limit is discontinuous.