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 2 of 9: Expand into initial terms

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a_n=a_{i_1}+\cdots+a_{i_t},\quad 1\le i_j\le s,\quad i_1+\cdots+i_t=n

Detailed analysis

Because every expansion lowers the expanded index, it terminates. Thus every $a_n$ can be represented as $a_n=a_{i_1}+\cdots+a_{i_t},\quad 1\le i_j\le s,\quad i_1+\cdots+i_t=n$ , with the indices at most $s$ ; the representation is chosen by selecting maximizing splits at each step.