Local finiteness is extension-closed

Statement with symbols
Suppose $$G$$ is a group with a normal subgroup $$H$$ such that both $$H$$ and $$G/H$$ are locally finite groups: any finitely generated subgroup of $$H$$ is finite, and any finitely generated subgroup of $$G/H$$ is finite. Then, $$G$$ is itself locally finite.

Related facts

 * Periodicity is extension-closed
 * Finite generation is extension-closed

Facts used

 * 1) uses::Schreier's lemma: In its factual form, this states that any subgroup of finite index in a finitely generated group is again finitely generated.
 * 2) uses::Finiteness is extension-closed
 * 3) uses::First isomorphism theorem

Proof
Given: A group $$G$$ with a normal subgroup $$H$$ such that both $$H$$ and $$G/H$$ are locally finite. Let $$\varphi:G \to G/H$$ be the quotient map.A finite subset $$A$$ of $$G$$ with $$K = \langle A \rangle$$.

To prove: $$K$$ is finite.

Proof: