Residual finiteness is subgroup-closed

This article gives the statement, and possibly proof, of a group property (i.e., residually finite group) satisfying a group metaproperty (i.e., subgroup-closed group property)
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Statement

Verbal statement

Any subgroup of a residually finite group is itself a residually finite group.

Statement with symbols

Suppose $G$ is a residually finite group and $H$ is a subgroup of $G$. Then, $H$ is also a residually finite group.

Facts used

1. Finiteness is subgroup-closed: Any subgroup of a finite group is finite.
2. Residually operator preserves subgroup-closedness: If $\alpha$ is a subgroup-closed group property, so is the property obtained by applying the residually operator to $\alpha$.