OIMU 2007 Problem 4

Consider an infinite sequence $a_1,a_2,\cdots$ whose terms all belong to $\left\{1,2\right\}$ . A positive integer with $n$ digits is said to be good if its decimal representation has the form $a_ra_{r+1}\cdots a_{r+(n-1)}$ , for some positive integer $r$ . Suppose that there are at least $2008$ good numbers with a million digits. Prove that there are at least $2008$ good numbers with $2007$ digits.