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 8 of 9: Translate stabilization back to the recurrence

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b_n=b_{n-\ell}+b_\ell=b_{n-\ell}\quad(n\ge N)

Detailed analysis

For $n\ge N$ , the equality $b_n=b_{n-\ell}+b_\ell=b_{n-\ell}\quad(n\ge N)$ says that the split using the distinguished index $\ell$ attains the maximum in the normalized recurrence.