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I was given the following problem:
Let $f:[0,1] \to \Bbb R $ be a differentiable function on $[0,1]$ such that $f(0)=0$, and $\forall x \in[0,1]$ : $\lvert f'(x)\rvert \le \lvert f(x)\rvert$. Prove that $f(x)=0$ $\forall x \in [0,1]$.
I tried to solve it using mean value theorem iteratively, and got to a point where I have a series that holds $f(x_1) \ge f(x_2) \ge f(x_3) \ge \dots $ pretty much stuck from here, is it a good start? any advice on how to move on / solve it differently? Thanks