The number $a=0.12457...$ is defined as follows: The digit on the $n$-th place after the dot is the first digit left to the dot of the number $n\sqrt2$.
For example, for $n=1$ we have
and its first digit to the dot is 1.
For $n=2$ we have
and its first digit left to the dot is 2.
For $n=3$ we have
and its first digit to the dot is 4.
I'd like to show that $a$ is irrational.