Free implies residually nilpotent

Verbal statement
Any free group is residually nilpotent group: the intersection of the terms of its finite lower central series is trivial.

Proof outline
Pick a freely generating set for the free group. Then, we show that the length of the shortest word for any non-identity element in the $$k^{th}$$ term of the lower central series, is bounded from below by $$k$$. Thus, if an element of the free group can be expressed as a word of length $$r$$, it cannot occur in the $$k^{th}$$ member for $$k > r$$.