Prove that if $f$ is entire and there exists a bounded sequence of distinct real numbers {$a_n$} with given property, show that $f$ is a constant. 
The above problem excerpted from Complex variables with application by Silverman, section 8.2.
The first part of the problem can be shown relatively easily by considering $g(z)=\overline{f(\bar{z})}$.
But I couldn’t show that the second part. I guess it probably can be proved by identity theorem or by comparing the coefficients of $f$.
Any help or hint will be appreciated.
Thanks.

Since $f(t) \in \mathbb R$ for all real $t$, there is, by Rolle, some $t_n \in (a_{2n+1}, a_{2n})$ such that $f'(t_n)=0.$
Since $t_n \to 0$, we see that $0$ is an accumulation point of the zeroes of $f'$. This gives that $f'(z)=0$ for all $z.$
