How to solve this exercise of Siegel's "Combinatorial Game Theory"?
Let $S_L$ and $S_R$ be sets of positive integers. The partisan subtraction game on $S_L$ and $S_R$ is played with a single heap of $n$ tokens. On her move, Left must remove $k$ tokens for some $k\in S_L$; likewise, on his move Right removes $k$ tokens for some $k\in S_R$. Denote by $o(H_n)$ the (normal play) outcome of a heap of $n$ tokens. Prove that the sequence $n \mapsto o(H_n)$ is periodic, i.e., there is some $p>0$ such that $o(H_{n+p})=o(H_n)$ for all sufficiently large $n$.
An important observation: this exercise is at the beginning of the book, so it is not supposed to use any advanced tool or theorem. I am probably being dumb and missing something obvious. My pride usually forbids me of ever giving up at solving a mathematical problem, but right now I really have to optimize my time, so I am asking for help instead of losing several days at this.
Thanks in advance.
Let $m$ be the largest element of $S_L\cup S_R$. Notice then that for $n\geq m$, $o(H_n)$ is completely determined by the $m$-tuple $$t_n=(o(H_{n-1}),o(H_{n-2}),\dots,o(H_{n-m})),$$ since this tells you the outcome of all possible positions you can reach after one move from a heap of $n$ tokens. There are only finitely many $m$-tuples of positions, and so eventually they must repeat: there exist $N'>N\geq m$ such that $t_N=t_{N'}$. This means that $o(H_N)=o(H_{N'})$, but then also $t_{N+1}=t_{N'+1}$ and so $o(H_{N+1})=o(H_{N'+1})$. It similarly follows by induction that $o(H_{N+k})=o(H_{N'+k})$ for all $k$, or in other words that $o(H_{n+p})=o(H_n)$ for all $n\geq N$ where $p=N'-N$.