I am having problems with this problem in my book. It says: If I have $n^2 +1$ distinct real numbers $a_1, a_2,...,a_{n^2+1}$. Assume that there is no increasing subsequence of length $n+1$. Let $j_i$ be the length of the longest increasing subsequence that starts at $a_i$. Prove that $1 \leq j_i \leq n$. I have no idea how to approach this problem. It is part of a bigger proof I am trying to do, but I am lost. Thank you for the help! :)
2026-03-30 03:54:55.1774842895
Prove the maximum length of a subsequence.
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