The theorem stats that
Every sequence n^2 + 1 distinct real numbers contains a subsequence of length n+1 that is either strictly increasing or strictly decreasing.
However, I came up with "counterexample", a sequence of numbers -- 5 6 4 7 3 8 2 9 1 10. It is a sequence of length (3^2 + 1) but there is not subsequence qualifying the theorem.
Not only do sequence members not need to be adjacent to qualify as a subsequence , but, strictly increasing or strictly decreasing mean $a>b$, $a$ being a number further up on the number line, and in the sequence, and $b<a$, $b$ being a number further down (as in towards the negative end) of the number line, respectively. Loosely (technically weakly), means they could be equal in value as well.