Thread: Question about the Uniqueness Theorem of Power Series

Let and be two convergent power series. Suppose that for all x in an infinite set having 0 as a point of accumulation. Then .

Proof.

Define , then 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.

Originally Posted by tttcomrader

Let and be two convergent power series. Suppose that for all x in an infinite set having 0 as a point of accumulation. Then .

Proof.

Define , then for an infinite set x having 0 as a point of accumulation. Now, I also know that if , 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 so that .

Suppose by contradiction that is not 0 everywhere. Then there is a least index such that , and we can factor the sum to write , so that you can see we have

This limit is also the limit along any sequence that converges to 0. However, you know there is such a sequence where is zero. We deduce from the above limit that , a contradiction.