I hope this will help...
Let be bounded sequence. Then exist and he the greater partial limit of the sequence.
In other words:
Due to Bolzano-Weierstrass Theorem we may discuss on the set of partial limits, which is non-empty.
Given there are infinite that , but only a finite number of . Therefor there is infinite in neighborhood of , hence is partial limit.
Now, suppose that other partial limit then we will choose so that , but we know that there is only finite number of , therefor it impossible that is partial limit, thus is the greatest partial limit.