Suppose ${f}_{n}$ is a sequence of $C([0,1])$, ${f'}_{n+1}={f}_{n}$ for all $n$ and:
$\forall x \in [0,1]$, $\exists n$ : ${f}_{n}(x)=0$.
I want to show that ${f}_{1}$ is identically null!
So I think we should use Baire Theorem like this: $E_{k}=${$x$ : ${f}_{k}(x)=0$}.
Then as $\bigcup_{n \geq 1} E_{n}$=$[0,1]$, there is an $E_{k}$ with non-empty interior, so $E_{k}$ contains an open interval! I don´t know where to go from here, would apreciate any help. Thanks in advance.