Observe that the sequence 12,15,8,13,7,18,19,11,14,10 has three increasing subsequences of length four: 12,15,18,19; 12,13,18,19; and 8,13,18,19. It also has one decreasing subsequence of length of distinct real numbers, there must be a sequence of length n+1 that is either strictly increasing or strictly or strictly decreasing.
To show that in any sequence of distinct real numbers, there must be a sequence of length that is either strictly increasing or strictly decreasing.
Observe the sequence has three increasing subsequence of length four; ; ; it also has one decreasing subsequence of length four: .
Let be any sequence of distinct numbers. Let us suppose an ordered pair for each where
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