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I need to prove that in any sequence of distinct numbers of length $n$ there is a monotonic subsequence of length at least $\lceil \sqrt{n} \rceil$.

I thought, that induction would probably be best here.

So $n=1$ and $n=2$ are trivial.

Now let's assume all is well for $k \leq n$ and consider $n+1$. If from the first $n$ elements we can create a decreasing/increasing sequence and the last elements is smaller/bigger than all of them then this is trivial. We can also do the same analysis with the first element.

But what about other cases? I have literally now idea how to approach them.

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  • $\begingroup$ Why "random"? Do you just mean any given sequence of $n$ numbers? $\endgroup$ – MPW Feb 8 '14 at 15:53
  • $\begingroup$ Yes. Sorry for the confusion. $\endgroup$ – Arek Krawczyk Feb 8 '14 at 15:55
  • $\begingroup$ What if the numbers just happen to be (1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2)? $\endgroup$ – MPW Feb 8 '14 at 15:55
  • $\begingroup$ Sorry again, added the requirement for the numbers to be distinct. $\endgroup$ – Arek Krawczyk Feb 8 '14 at 15:56
  • $\begingroup$ Oh, I guess you don't mean in the order given. So you could rearrange my numbers as 1,1,1,1,1,1,1,...,2,2,2,2,2,2,2 . I see. $\endgroup$ – MPW Feb 8 '14 at 15:56
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This is the theorem of Erdos and Szekeres. There are about a half dozen proofs in the paper http://stat.wharton.upenn.edu/~steele/Papers/PDF/VOTMSTOEAS.pdf

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