Let $M=\max \lbrace \text{sum of digits of }n:0\leq n\leq [a]\rbrace$, and for each $k\in \mathbb{N}$ let $n_k$ be the smallest integer such that $[an_k+b]\geq 10^k$. Then $10^k\leq an_k+b<10^k+a$, so $x_{n_k}\leq M+1$ for all $k$, hence by the pigeonhole principle infinitely many elements from the sequence $\lbrace x_{n_k}\rbrace$ are equal.

Consider $n_k=[\frac{10^k}{a}]+1$. Obviosly $x_{n_k}$ bounded, therefore there is constant subsequence $x_{n_k}$.