Question 7
Let $a_{1}, \ldots, a_{r}$ be positive real numbers. For $n\gt r$, we inductively define
$$
a_{n}=\max _{1 \leq k \leq n-1}\left(a_{k}+a_{n-k}\right)
$$
Prove that there exist positive integers $\ell \leq r$ and $N$ such that $a_{n}=a_{n-\ell}+a_{\ell}$ for all $n \geq N$.
$$
a_{n}=\max _{1 \leq k \leq n-1}\left(a_{k}+a_{n-k}\right)
$$
Prove that there exist positive integers $\ell \leq r$ and $N$ such that $a_{n}=a_{n-\ell}+a_{\ell}$ for all $n \geq N$.