Set $B_n = (B_n,\ldots,a_{n+m-1})$ for $n \ge 1$. The sequence $(B_n)_{n \ge 1}$ thus defined takes values in a finite set (namely $\{0,\ldots,9\}^m$) and satisfies a recursion relation of the form $B_{n+1}=f(B_n)$. Therefore, it is eventually periodic: once you get any value for the second time, the periodic behaviour is forced. The linearity plays no role here.

Set $B_n = (a_n,\ldots,a_{n+m-1})$ for $n \ge 1$. The sequence $(B_n)_{n \ge 1}$ thus defined takes values in a finite set (namely $\{0,\ldots,9\}^m$) and satisfies a recursion relation of the form $B_{n+1}=f(B_n)$. Therefore, it is eventually periodic: once you get any value for the second time, the periodic behaviour is forced. The linearity plays no role here.