If H is a subgroup of K, K is a subgroup of G, then |G:H| = |G:K||K:H| (do not assume G is finite).
Let be a left coset where . We know that for some since is a complete collection of cosets. Therefore for some . Now for some since is a complete collection of cosets. Therefore, for some . Thus, . We have shown that contains all cosets of in . Therefore, there are at most cosets of in . The reason why at most and not exactly is because it is possible that different 's and 's can produce the same coset. It remains to show that if then . Can you finish the last step?