Problem 6

Let

a_1,a_2,a_3,\ldots

be a sequence of positive real numbers. Suppose that for some positive integer

s

, we have

a_n=\max\{a_k+a_{n-k}\mid 1\le k\le n-1\}

for all

n>s

. Prove that there exist positive integers

\ell

and

N

, with

\ell\le s

, such that

a_n=a_\ell+a_{n-\ell}

for all

n\ge N

Step 3 of 9: Choose the maximal initial slope

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q=\max_{1\le i\le s}\frac{a_i}{i},\qquad a_\ell/\ell=q

Detailed analysis

Set $q=\max_{1\le i\le s}\frac{a_i}{i},\qquad a_\ell/\ell=q$ for an index $\ell$ attaining the maximum. Then $1\le\ell\le s$ and $a_i\le qi$ for every $1\le i\le s$ .