I'll copy and paste the background information in my other question:
Given a filtration of complexes $\emptyset = K_0 \subseteq K_1 \subseteq \ldots \subseteq K_n = K$, we have induced homomorphisms $h^{i,j}_p \colon H_p(K_i) \to H_p(K_j)$ for $i \leq j$.
The pth persitent homology groups are the images of $h^{i,j}_p$.
And we say that class $\phi \in H_p(K_i)$ is born at $K_i$ if $\phi \notin H^{i-1,i}_p$. And class born at $K_i$ dies entering $K_j$ if $h^{i, j-1}_p(\phi) \notin H_p^{i-1,j-1}$ but $h_p^{i,j}(\phi) \in H^{i-1,j}_p$
Now we have the following:
where $\beta^{i,j}_p$ is the rank of $H^{i,j}_p$, the image of $h^{i,j}_p$.
The text says that $\mu^{i,j}_p$ is the number of classes born at $H_p(K_i)$ that dies at $H_p(K_j)$.
And $\beta^{i, j-1}_p - \beta^{i,j}_p$ is supposed to count the number of classes born at or before $K_{i-1}$ and die entering $X_j$.
What I don't get is that $\beta^{i, j-1}_p - \beta^{i,j}_p$ is the number of classes born at or before $X_i$ and die entering $X_j$.
And since LHS is just the same thing as RHS with $i$ replaced with $i-1$, I would be done if I understood the above.
Thank you.
First, we need to know this:
Similarly, $\beta_p^{i,j-1}$ counts the homology classes in $X_i$ that are still alive in $X_{j-1}$.
Hence, their difference $\beta_p^{i,j-1}-\beta_p^{i,j}$ counts the homology classes in $X_i$ that are still alive in $X_{j-1}$, but no longer alive in $X_j$.
In other words, those that are born at or before $X_i$ and die entering $X_j$.
A good reference is this set of notes, pg. 130.