It doesn't say you need both directions. It just says you need a monotonic subsequence. Clearly if you have a monotonic increasing sequence it's not going to have a monotonic decreasing subsequence and vice versa.
I need help with understanding Lemma,
Lemma: For every sequence (an) in R it has monotonic subsequence.
Firstly, why is this Lemma true?I mean you can have a sequence in R that is monotonically increasing, but I can't find a subsequence that is monotonically decreasing. I can find a subsequence that is monotonic in the same direction but not other directions.. ( By direction I mean increasing and decreasing)
I think I didn't understand the Lemma can someone explain? Thanks