1. ## Question about the Uniqueness Theorem of Power Series

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

Proof.

Define $h=f-g= \sum (a_n=b_n)T^n$, then $h(x)=0$ 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.

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

Proof.

Define $h=f-g= \sum (a_n-b_n)T^n$, then $h(x)=0$ for an infinite set x having 0 as a point of accumulation. Now, I also know that if $h(0){\color{red}\neq} 0$, 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.
(note the inequality in red; I guess this is what you meant?)

Let $c_n=a_n-b_n$ so that $h(x)=\sum_{n=0}^\infty c_n x^n$.

Suppose by contradiction that $h$ is not 0 everywhere. Then there is a least index $n_0\geq 0$ such that $c_{n_0}\neq 0$, and we can factor the sum to write $h(x)=x^{n_0}\sum_{n=0}^\infty c_{n_0+n} x^n=x^{n_0} \left(c_{n_0}+c_{n_0+1}x+\cdots\right)$, so that you can see we have

$c_{n_0}=\lim_{x\to0^+}\frac{h(x)}{x^{n_0}}$

This limit is also the limit along any sequence $(x_n)_n$ that converges to 0. However, you know there is such a sequence where $h$ is zero. We deduce from the above limit that $c_{n_0}=0$, a contradiction.