PAMO 2010 Problem 5

A sequence $a_0,a_1,a_2,\ldots ,a_n,\ldots$ of positive integers is constructed as follows:

• if the last digit of $a_n$ is less than or equal to $5$ then this digit is deleted and $a_{n+1}$ is the number consisting of the remaining digits. (If $a_{n+1}$ contains no digits the process stops.)
• otherwise $a_{n+1}=9a_n$ .

Can one choose $a_0$ so that an infinite sequence is obtained?