TITLE: How to prove that every real number is a zero of some power series with rational coefficients (if true)
QUESTION [17 upvotes]: How would one approach proving that every real number is a zero of some power series with rational coefficients? I suspect that it is true, but there may exist some zero of a non-analytic function that is not a zero of any analytic function. I was thinking about approaching the problem using arguments of cardinality, but I am unsure about how to begin.
Thank you in advance.

REPLY [4 votes]: From "Note on integral functions" by A.G Walker:

The following theorem, though perhaps well known, does not appear in any of the elementary text-books of analysis. It may, however, be of interest in connection with the various rational representations of irrational numbers, and an elementary proof is given.
Theorem. Every real number is a zero of a rational integral function, i.e. a function expressible as a power series, with rational coefficients, convergent for all x.

The proof is basically the same as the one in Gerry's answer.