Show $H_n(X_n,\,X_{n-1}) \cong H_n(X_n/X_{n-1}, \ast)$ Assume $ X_{-1}:=\emptyset, X_0 \subset X_1 \subset \dots \subset X\;$ is a CW-complex with pushouts
$$
\newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex}
\newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex}
\newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}}
\begin{array}{c}
%
\bigsqcup_{i \in I} S^{n-1} & \ra{\hspace{0.35cm} <\varphi_i>_i \hspace{0.35cm}} & X_{n-1} \\
\da{inc} &  & \da{inc}\\
\bigsqcup_{i \in I}  D^{n} & \ras{<\phi_i>_i} & X_{n}\\
\end{array}\quad,\;n \in \mathbb{N} \\
%
$$
where '$inc$' denotes the set-inclusion in all cases.
How do I show that
$$
H_n(X_n,\,X_{n-1}) \cong H_n(X_n/X_{n-1}, \ast)
$$
is an isomorphism of groups? Note that $H_n$ denotes the $n$-th functor of singular homology. Also, the $\ast$ is shorthand for $\{X_{n-1}\}$, i.e. the class of $X_{n-1}$ under the canonical projection.
I tried using the LESes for the respective pairs, connecting them through the projection-induced morphisms. But I don't think that was enough...
I am studying for an exam and would appreciate if someone could help me with this, complete answers are appreciated. This isomorphy is part of a proof in the lecture notes, but the professor didn't bother to make a remark about it.
 A: As Kenny Wong pointed out, this follows, because $(X_n, X_{n-1})$ is a good pair. Stacking pushouts left to right,
$$
\newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex}
\newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex}
\newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}}
\begin{array}{c}
%
\bigsqcup_{i \in I} S^{n-1} & \ra{\hspace{0.35cm} <\varphi_i>_i \hspace{0.35cm}} & X_{n-1} & \ra{\hspace{1.3cm}} & \ast \\
\da{inc} &  & \da{inc} & & \da{}\\
\bigsqcup_{i \in I}  D^{n} & \ras{<\phi_i>_i} & X_{n} & \ras{\hspace{0.9cm}} & X_n/X_{n-1}\\
\end{array}\quad,\;n \in \mathbb{N} \\
%
$$
use that, because we have these pushouts, we can apply the following theorem from left to right:
Given a homology theory $h_*$(for instance, singular homology $H_*$) and a pushout of topological spaces
$$
\newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex}
\newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex}
\newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}}
\begin{array}{c}
%
A & \ra{\hspace{0.9cm}} & X \\
\da{} &  & \da{}\\
B & \ras{\hspace{0.9cm}} & Y\\
\end{array}\\
%
$$
where $(B,\,A)$ is a good pair, then $(Y,X)$ is a good pair and homology induces isomorphisms $h_n(B,\,A) \cong h_n(Y,\,X)\,,\;n \in \mathbb{Z}$.
