Let be a sequence.

If :

Then is called a peak point of the sequence.

Every sequence has a Monotonic Sequences (subsequence)

Proof
Let be the collection of all peak points.

Case 1:
is infinite

The collection of forms an increasing infinite sequence. By definition they are increasing and therefore monotonic

Case 2:
is finite

Let their exist an which is larger than any peak point.

Similarly we can say given that because otherwise must be a peak point.

By induction we can define an infinite non decreasing (monotonic) sequence.