Normality satisfies lower central series condition

Statement
Suppose $$G$$ is a group and $$H$$ is a normal subgroup of $$G$$. Suppose $$k$$ is a positive integer. Denote by $$\gamma_k(G)$$ the $$k^{th}$$ member of the lower central series of $$G$$, and denote by $$\gamma_k(H)$$ the $$k^{th}$$ member of the lower central series of $$H$$. Then, $$\gamma_k(H)$$ is a normal subgroup of $$\gamma_k(G)$$.

Facts used

 * 1) uses::Lower central series member functions are monotone, i.e., if $$H \le G$$, then $$\gamma_k(H) \le \gamma_k(G)$$.
 * 2) uses::Normality is preserved under any monotone subgroup-defining function

Proof
The proof follows directly by combining Facts (1) and (2).