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 9 of 9: Conclusion

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a_n=b_n+qn=a_{n-\ell}+q\ell=a_{n-\ell}+a_\ell

Detailed analysis

Finally, $a_n=b_n+qn=a_{n-\ell}+q\ell=a_{n-\ell}+a_\ell$ for all sufficiently large $n$ . Since $1\le\ell\le s$ , this is exactly the required assertion.