The question is given below:
My questions are:
1-I do not understand the part of the question starting from the word hence (the first part I have already proved it), could anyone explain this part for me please?
2- A stupid question, do we have to prove the part starting from the word hence?
The first line refers to a result in general for Abelian categories.
It is sufficient for you to work with the category for abelian groups. You may attempt the general case as an exercise. For starters, you need to know:
(a) What it means to be exact.
(b) What it means to be injective and surjective.
Now I will prove one part of the first line.
$A \rightarrow B$ is surjective iff $im (A \rightarrow B) = B$ iff $\ker (B \rightarrow C)= B$ iff im $(B \rightarrow C) =0$
$D \rightarrow E$ is inejctive iff $\ker (D \rightarrow E)=0$ iff $im(C \rightarrow D) =0$ iff $\ker(C \rightarrow D)=C$.
Now result follows if you know what (a) means.
For the second part,
Apply the first to the LES in homology groupos induced from the SES,
$$ C_*(A) \rightarrow C_*(X) \rightarrow C_*(X)/C_*(A) $$ In which case, we have $$\cdots \rightarrow H_n(A) \rightarrow H_n(X) \rightarrow H_n(X,A) \rightarrow H_{n-1}(A) \rightarrow \cdots $$ If $H_n(X,A)=0$ , then by first part $H_n(A) \rightarrow H_n(X)$ is an isomorphism.