Yes, but that is irrelevant to the question. Any such convergent subsequence is "eventually constant". The definition of "cluster point" is that p is a cluster point if, given any
, there exist an
infinite] number of points
other than p itself whose distance to p is less than
. If the sequence has only a finite number of points then the set of distances a given point p and points in the sequence, other than p itself, is finite and has a non-zero minimum. Let [itex]\epsilon[/itex] be that minimum distance. Given any p, there exist
no other point whose distance is less than
. A finite set does NOT have a cluster point and so no finite valued sequence has a cluster point.