Hatcher 4.21 says that a weak homotopy equivalence $f : X\to Y$ induces isomorphisms $f_* : H_n(X; G)\to H_n(Y; G)$.
In the proof he says that by replacing $Y$ by the mapping cylinder $M_f$ and looking at the long exact sequences of homotopy, homology, and cohomology groups for $(M_f, X)$, we see that it suffices to show:__ If $(Z, X)$ is an n connected pair of path-connected spaces, then $H_i(Z, X; G) = 0$ for all $i \leq n$ and all $G$.
Question: How can we see this from looking at the long exact sequences? Additionally, why do we stipulate that $(Z,X)$ is an n connected pair?
I'm a little bit lost as to why this should be enough to prove the statement. Any help would be appreciated.
Given a pair of spaces like $(M_f, X)$, there are long exact sequences in homotopy and homology. $M_f$ and $Y$ are homotopy equivalent, so $X \to M_f$ is a weak homotopy equivalence. Therefore from the long exact sequence in homotopy $$ \dots \to \pi_m(X) \to \pi_m(M_f) \to \pi_m(M_f, X) \to \pi_{m-1}(X) \to \dots, $$ we see that $\pi_m(M_f, X)=0$ for all $m$. Therefore $(M_f, X)$ is $n$-connected for any $n$. So if we can conclude, as Hatcher proposes, that $H_i(M_f, X; G)=0$ for all $i \leq n$, then from the long exact sequence in $H_{*}(-;G)$, we will get isomorphisms $H_i(X;G) \to H_i(M_f;G)$ for all $i < n$.