How would I go about proving such a statement? Any help would be great thanks.
First prove this theorem: Every sequence contains a monotone subsequence.
Then prove that Every bounded non-increasing (non-decreasing) sequence converges to its least upper bound (greatest lower bound).
Suppose that is a sequence.
Define a set .
Take notice of this critical fact: .
There are two cases to consider: is infinite.
In this case there is an increasing subsequence.
If is finite then there is a non-increasing subsequence.
Well, that would be trivial! Just add some numbers to your monotone sequence! For example, the monotone sequence is a subsequence of sequence
But what you want is the other way- that every sequence contains a monontone subsequence. I'll give you a start: Let S be the set of all indices i such that if j> i, then . In other words, i is in S if and only if is less than all succeeding numbers in the sequence. Now, consider two cases:
1) S is infinite.
2) S is either finite or empty.