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 4 of 9: Subtract the linear part

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b_n=a_n-qn,\qquad b_\ell=0,\qquad b_i\le0\ (1\le i\le s)

Detailed analysis

Define $b_n=a_n-qn$ . The choice of $q$ gives $b_\ell=0,\qquad b_i\le0\ (1\le i\le s)$ . Applying the expansion from the preceding step shows $b_n\le0$ for every $n$ : each expansion is a sum of terms $b_{i_j}$ , and all its initial terms are non-positive.