Free implies residually nilpotent

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property must also satisfy the second group property
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Statement

Verbal statement

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

Definitions used

Free group

Further information: free group

Residually nilpotent group

Further information: residually nilpotent group

Proof

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 $k>r$.