Need to prove that $f=0$ given that f is an entire function and $|f(1/n)|<\frac{1}{2^n}$ for all n in natural numbers.

Question

I've seen that, as f is an entire function, I can write its Taylor series expansion around z=0 and prove that $f^{(n)}(0)=0$ for all n but, I 've got a lot of problems when trying to prove it by induction. I don't know if there is an easier way to prove this fact or maybe see the problem from a different point of view.