Suppose that power series function $a_0 + a_1x + a_2x^2 + a_3x^3 + ...$ is constantly zero on a bounded non-empty open interval $I$, which may or may not contain $0$.
Prove that $a_j = 0$ for every $j$, so that the power series is constantly zero on its whole domain (which is of course centered at $0$).
Here is my thought: let us call power series function given in the problem $f$. Let $I = (a, b)$ (where $a < b$, of course), and let $c= (a+b)/2=$ the center of $I$. I don't know how to go from here. Any help?