Question 6
Let $a_1, a_2, a_3, \dots$ be a sequence of positive real numbers, and $s$ be a positive integer, such that $$
a_n = \max \{ a_k + a_{n-k} \mid 1 \leq k \leq n-1 \} \text{ for all } n \gt s.
$$ Prove there exist positive integers $\ell \leq s$ and $N$, such that $$
a_n = a_{\ell} + a_{n - \ell} \text{ for all } n \ge N.
$$
a_n = \max \{ a_k + a_{n-k} \mid 1 \leq k \leq n-1 \} \text{ for all } n \gt s.
$$ Prove there exist positive integers $\ell \leq s$ and $N$, such that $$
a_n = a_{\ell} + a_{n - \ell} \text{ for all } n \ge N.
$$