# Prove that for a power series function that is constantly zero, that the coefficients are zero

Suppose that power series function $a_0 + a_1x + a_2x^2 + a_3x^3 + \cdots$ 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 $0$).