Firstly, i'm sorry for getting stuck three times in one day! I'm pretty much hogging this board which isn't that nice.

Anyway, i'm stuck on this:

Kindly (?), the homework sheet offers some hints in the form of more questions!Prove that every sequence has a monotonic subsequence.

My answer to 1).1). If there are an infinite number of floor terms, show they form a monotonic increasing subsequence.

2).If there are a finite number of floor terms and the last one is , construct a monotonic decreasing subsequence with as it's first term.

3). If there are no floor terms, construct a monotonic decreasing subsequence with as it's first term.

Since (ie. the definition of a floor term) then let

Therefore there are an infinite number of floor terms and the next floor term is greater than the previous one.

My answer to 2).

Starting once more from the definition . is the first term and is the last term.

Let and .

Therefore

Therefore there is a monotonic decreasing subsequence as required.

(I think so anyway. Could someone double check this?)

My answer to 3).

I decided to choose since this doesn't have a lower bound. Would floor terms be in the sequence ?

Therefore making f=1:

etc

However, this doesn't create a monotonic decreasing subsequence and I can't think of a sequence that does!

Could I have some help with these "hints" so I can use them to solve the question?

Thank you kindly to anyone who posts!