Equivalence of definitions of nilpotent group

Definition in terms of central series
A group is nilpotent if it possesses a central series.

Definition in terms of upper central series
A group is nilpotent if its upper central series stabilizes after a finite length, at the whole group.

Definition in terms of lower central series
A group is nilpotent if its lower central series stabilizes after a finite length, at the trivial subgroup.

Facts used

 * Lower central series is fastest descending central series
 * Upper central series is fastest ascending central series

Proof
We'll show:

Upper central series definition $$\iff$$ Central series definition $$\iff$$ Lower central series definition

From upper central series to central series
If the upper central series has finite length, it gives a central series for the group.

From lower central series to central series
If the lower central series has finite length, it gives a central series for the group.

From central series to upper central series
By the fact that upper central series is fastest ascending central series, we can prove that the upper central series must stabilize at the whole group after a finite length, and that this length is bounded from above by the length of the central series we started with.

From central series to lower central series
By the fact that lower central series is fastest descending central series, we can prove that the upper central series must stabilize at the trivial subgroup after a finite length, and that this length is bounded from above by the length of the central series we started with.

Incidentally, a slight modification of this proof yields the equivalence of definitions of nilpotence class.