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?