MathDB

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)

Problem Statement