Proof That A Power Series Only Equals Zero If and Only If Its Coefficients Equal Zero

Question

I've heard it said that a power series equals zero iff all of its coefficients equal zero.

That is:

$$\sum_{n=0}^\infty a_n\, x^n=0\quad\iff\quad a_i =0\quad \forall i \;\in \Bbb{N}$$