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! :)
Prove the maximum length of a subsequence.
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real-analysis
sequences-and-series
pigeonhole-principle
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0If your $j_i$ is $\geq n+1$, then you can pick an increasing subsequence of length $n+1$ from this sequence. – 2017-02-13
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0So just contradiction and that's it? – 2017-02-13
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0I dont know, it seems so, but I suggest you to check your problem again, it shouldn't be like this simple. – 2017-02-13
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0That is literally all that appears in the book for that part of the problem, haha. – 2017-02-13