MathNet 2014 Question 27

For a positive integer $x$, let $S(x)$ be its decimal digit sum. Let $k\gt 0$ be an integer and define $x_1=1$, $x_{n+1}=S(kx_n)$ for every $n\gt 0$. Prove that $x_n\lt 27\sqrt{k}$ for all $n\gt 0$.