irrationality of a decimal expansion

Question

Consider the real number in $(0,1)$ having the decimal expansion $${\alpha} = 0.{a_1}{a_2}{a_3}\cdots $$ where $a_j$ is obtained by adding up the digits in the decimal expansion of the positive integer $j$, and then reducing the sum $({\rm mod}\ 10).$ Thus, we can write the first few entries in the decimal expansion as follows: $$0.123456789123456789023456789\cdots$$ Prove that $\alpha$ is an irrational number.