RU-ALLRUSSIAN 1991 Problem 17

A sequence of positive integers is constructed as follows. If the last digit of $\mathbf{a}_{\mathrm{n}}$ is greater than 5, then $\mathbf{a}_{\mathrm{n} + 1}$ is $9\mathbf{a}_{\mathrm{n}}$ . If the last digit of $\mathbf{a}_{\mathrm{n}}$ is 5 or less and $\mathbf{a}_{\mathrm{n}}$ has more than one digit, then $\mathbf{a}_{\mathrm{n} + 1}$ is obtained from $\mathbf{a}_{\mathrm{n}}$ by deleting the last digit. If $\mathbf{a}_{\mathrm{n}}$ has only one digit, which is 5 or less, then the sequence terminates. Can we choose the first member of the sequence so that it does not terminate?