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
$n\sqrt2=\sqrt2=1.4142...$
and its first digit to the dot is 1.
For $n=2$ we have
$n\sqrt2=2\sqrt2=2.8284...$
and its first digit left to the dot is 2.
For $n=3$ we have
$n\sqrt2=3\sqrt2=4.2426...$
and its first digit to the dot is 4.
I'd like to show that $a$ is irrational.
Consider the cases $n=10^k$. Then we get that the $n$th digit of $a$ is the $k$th digit of $\sqrt{2}$. Now, if $a$ is rational, then it repeats with some frequency, $f$. But then we can find $d$ so that $f\mid 10^{k+d}-10^k$ for $k$ large enough. Therefore, for large enough $k$, the $10^{k+d}$th digit of $a$ and the $10^k$th digit of $a$ must be the same.
But that means that $k+d$th digit of $\sqrt{2}$ is the same as the $k$th digit of $\sqrt{2}$ for large enough $k$, and therefore $\sqrt{2}$ repeats, and therefore $\sqrt{2}$ is rational, which is a contradiction.