Need help to understand a lemma

I need help with understanding Lemma,

Lemma: For every sequence (a_{n}) 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

Re: Need help to understand a lemma

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.

Re: Need help to understand a lemma