Frankly, I follow little of what you posted. Why are you doing so may different things.

Look carefully at what you are asked to show.

Suppose that is a a monotone sequence.

Without loss of generality (wlog) we can assume the sequence is non-decreasing.

For proof by contradiction, suppose the sequence has two cluster points, .

Again wlog we can assume . Let .

The open intervals .

By definition of cluster point, each of those intervals must contain infinitely many points of .

Is the possible for a non-decreasing sequence?