A short exact sequence

Question

I have this proposition, and I don't understand how to do to obtain the short exact sequence:

where axiom 4 is:

Answer 1

Because $i_*$ is injective, the images of the boundary maps $\partial$ are zero (by exactness of the sequence given in axiom 4). The exactness of the sequence also means that $j_*$ is surjective. Hence the long exact sequence given by axiom 4 breaks up into short exact sequences

$$0\overset{\partial}{\to} H_k(A)\overset{i_*}{\to}H_k(X)\overset{j_*}{\to}H_k(X,A)\overset{\partial}{\to}0$$