Let b1 b2 b3 etc be a set of right cosreps of H in G, one representative for each coset. Let ci be the inverse of bi.
For convenience, let b1, the representative of H itself, equal 1, the identity element. Thus c1 is also equal to 1.
We would like to build a set of generators for H, using the generators of G and the coset representatives. This process is called Schreier (biography) Nielsen (biography) rewriting.
For each i and j, build an element vi,j = biajck. In each case, choose k so that the constructed element winds up in H. In other words, bk represents the same coset as biaj.
The inverse of vi,j applies a cosrep, then the inverse of aj, then another cosrep, pulling the word back into H.
Everything in G, and in H, is spanned by the generators of G. Assume a word, built from these generators, lies in H. If the first symbol is a7, replace it with v1,7. This leaves some ck at the end. if the next symbol is a3, replace it with vk,3. The adjacent letters ckbk will cancel, leaving a7a3, as they appeared in the original word. Continue this process, replacing each symbol with one of our new generators vi,j, or its inverse if the symbol is inverted. When the word is rewritten, the last ck has to be 1, because the word lies in H.
Each generator lies in H, and every word in H has been spanned. We have constructed a set of generators for the subgroup H.
If G is finitely generated and H has finite index in G, H is finitely generated.