Normality satisfies lower central series condition

This article gives the statement, and possibly proof, of a subgroup property (i.e., normal subgroup) satisfying a subgroup metaproperty (i.e., lower central series condition)
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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. Lower central series member functions are monotone, i.e., if $H \le G$, then $\gamma_k(H) \le \gamma_k(G)$.
2. Normality is preserved under any monotone subgroup-defining function

Proof

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