shaoyi he wrote: > >in discrete mathematics and its applications 6th ,in Pigeonhole >Principle, the author give a THEOREM(page 351): > >Every sequence of n^2 + 1 distinct real numbers contains >a subsequence of length n + 1 that is either strictly increasing >or strictly decreasing. > >i donnot know what's the theorem for? because when we sort the >n^2 + 1 distinct real numbers, we can get n^2+1 that is either >strictly increasing or strictly decreasing. so how to >understand this?
The theorem asserts that without changing the order of the terms, there is already a strictly increasing or strictly decreasing subsequence of length n + 1.