The long exact sequence has the following form

(Hatcher p 115).

Since A has at least as many path-components as X, is surjective (They are both free abelian groups and H_0(A) has the same or higher rank than H_0(X)).

Let and . By the definition of the long exact sequence, .

Since is surjective, it forces N is a zero map and (The kernel of N should be the whole image of M, which is H_0(X)) .