Let $u_0,u_1,\ldots $ be a sequence of positive integers such that $u_0$ is arbitrary and for any non-negative integer $n$ $$u_{n+1}=\begin{cases}\frac{1}{2}u_n & \text{for even }u_n, \\ a+u_n & \text{for odd }u_n, \end{cases}$$ where $a$ is some fixed odd positive integer. Prove that the sequence is periodic from a certain step.