Problem

Source: Ukraine TST 2012 p6

Tags: irrational number, irrational, Digits, decimal representation, algebra



For the positive integer $k$ we denote by the $a_n$ , the $k$ from the left digit in the decimal notation of the number $2^n$ ($a_n = 0$ if in the notation of the number $2^n$ less than the digits). Consider the infinite decimal fraction $a = \overline{0, a_1a_2a_3...}$. Prove that the number $a$ is irrational.