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 n 2 + 1 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 n2+1 distinct real numbers, there must be a sequence of length n+1 that is either strictly increasing or strictly decreasing.

Given information:

Observe the sequence 12,15,8,13,7,18,19,11,14,10 has three increasing subsequence of length four; 12,15,18,19 ; 12,13,18,19 ; 8,13,18,19 it also has one decreasing subsequence of length four: 15,13,11,10.

Calculation:

Let S={a1,a2,a3,.......,an2+1} be any sequence of n2+1 distinct numbers. Let us suppose an ordered pair (mr,nr) for each ar,1≤r≤n2+1 where mr</