Density of zeros of a power series over the reals Let $f(x) = \sum_{i =0}^\infty a_i x^i$ be a power series which converges for all real $x$. Assume that $f(x)$ is not identically zero. I'm interested in the density of the zeros of $f(x)$. Let $Z$ be the set of zeros of $f(x)$. Which of the following claims about density of $Z$ are true? 
Claim 1: $Z$ is nowhere dense.
Claim 2: $Z$ is countable.
Claim 3: For any $a,b \in \mathbb{R}$ , $Z \cap [a,b]$ is finite.
I believe (correct me if I'm wrong) that claim 3 implies the other two. I suspect all three claims are true.
I suspect that the answers to these questions are well-known, though I was not able to find an obvious reference. Can anyone suggest a reference with a nice treatment of these questions?

All three claims are true.  Claim 3 $\Rightarrow$ Claim 2 and Claim 1. The zeros of an analytic function are isolated. This implies Claim 3.
