Question about the Uniqueness Theorem of Power Series

Let $\displaystyle f(T)= \sum a_nT^n$ and $\displaystyle g(T)= \sum b_nT^n$ be two convergent power series. Suppose that $\displaystyle f(x)=g(x)$ for all x in an infinite set having 0 as a point of accumulation. Then $\displaystyle f(T)=g(T)$.

Proof.

Define $\displaystyle h=f-g= \sum (a_n=b_n)T^n$, then $\displaystyle h(x)=0$ for an infinite set x having 0 as a point of accumulation. Now, I also know that if h(0)=0, then there exist a neighborhood centered at 0 such that h is non-constant (I actually understand the proof of that).

But how do I incorporate this idea into the proof to arrive at the conclusion that h is 0 everywhere? Thank you.