TITLE: subgroups of finitely generated groups with a finite index
QUESTION [35 upvotes]: Let $G$ be a finitely generated group and $H$ a subgroup of $G$. If the index of $H$ in $G$ is finite, show that $H$ is also finitely generated.

REPLY [6 votes]: If $G$ is finitely generated by $A\subset G$, equipping $G$ with the word metric, we can look at $G$ as a proper metric space $(G,d_{A})$. Then $H$ acts continuously (by isometries) on $(G,d_{A})$. Provoking Švarc–Milnor lemma, we get that $H$ is f.g and quasi-isometric to $G$ (looks like $G$ looking at both $G$ and $H$ from far distance).