A sequence $a_0, a_1, \dots , a_n, \dots$ 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 this sequence is infinite?