Let $(x_n)_{n> 0}$ a sequence of $\{ 0,1 \}$ and
$$ x=\sum_{n=1}^{\infty}\frac{x_n}{10^n}. $$
Prove that if $x$ is irrational then $x$ is transcendental.
I tried to first start by proving that $x$ can't be the root of a second degree polynomial with integer coefficients, but I couldn't do it.