Finite index in finitely generated implies finitely generated

Statement
Any subgroup of finite index in a finitely generated group, is finitely generated.

Explanation
In general, a subgroup of a finitely generated group need not be finitely generated. The problem in general is that the subgroup could be very deep inside the group, and thus it may not resemble the whole group at all. If, however, the subgroup is sufficiently close to the whole group, in the sense of having finite index, then we can use a generating set for the whole group to obtain a generating set for the subgroup. In general, the higher the index of the subgroup, the more the number of generators we may need.

A result of similar flavour in commutative algebra is the Artin-Tate lemma.

Proof
The proof is a direct application of Schreier's lemma.

Suppose $$G$$ is a finitely generated group with finite generating set $$A$$, and $$H$$, is a subgroup. Suppose $$S$$ is a left transversal for $$H$$ in $$G$$, i.e. a system of left coset representatives. Since $$[G:H]$$ is finite, $$S$$ is a finite set.

Schreier's lemma then constructs for us a generating set for $$H$$, whose cardinality is given by the product of the cardinalities of $$S$$ and $$A$$. Thus, $$H$$ is finitely generated, and moreover, an upper bound on the size of a smallest generating set for it is given by a product of its index, and the size of a smallest generating set for $$G$$.