I need help with understanding Lemma,

Lemma: For every sequence (a_{n}) init has monotonic subsequence.R

Firstly, why is this Lemma true?I mean you can have a sequence inthat 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)R

I think I didn't understand the Lemma can someone explain? Thanks