First prove this theorem:Every sequence contains a monotone subsequence.

Then prove thatEvery 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.