IMC 2014 Problem 6

For a positive integer $x$ , denote its $n^{\mathrm{th}}$ decimal digit by $d_n(x)$ , i.e. $d_n(x)\in \{ 0,1, \dots, 9\}$ and $x=\sum_{n=1}^{\infty} d_n(x)10^{n-1}$ . Suppose that for some sequence $(a_n)_{n=1}^{\infty}$ , there are only finitely many zeros in the sequence $(d_n(a_n))_{n=1}^{\infty}$ . Prove that there are infinitely many positive integers that do not occur in the sequence $(a_n)_{n=1}^{\infty}$ .

(Proposed by Alexander Bolbot, State University, Novosibirsk)