This question stems from page 28 of the "Morse Inequalities" chapter of Milnor's Morse Theory.
Given a pair of topological spaces, $Y \subset X$ and any field $F$, define $R_\lambda(X,Y)$ to be the rank over $F$ of $H_\lambda(X,Y;F)$ (also called the $\lambda$th Betti number of the pair $(X,Y)$). According to Milnor, the Betti number is subadditive in the following sense. Given the triple of spaces $Z \subset Y \subset X$, then $R_\lambda(X,Z) \leq R_\lambda(X,Y) + R_\lambda(Y,Z)$.
Milnor says the subadditivity follows from the exact homology sequence for a triple: $\begin{align*} \cdots \rightarrow H_\lambda(Y,Z) \rightarrow H_\lambda(X,Z) \rightarrow H_\lambda(X,Y) \rightarrow \cdots \end{align*}$ But I don't see how it follows. Could someone explain?
From the long exact sequence one obtains a short exact sequence $$1 \to \underbrace{\text{image}\bigl( H_\lambda(Y,Z) \to H_\lambda(X,Z)\bigr)}_{K=\text{cokernel}\bigl(H_{\lambda+1}(X,Y) \to H_\lambda(Y,Z)\bigr)} \to H_\lambda(X,Z) \to \underbrace{\text{image}\bigl(H_\lambda(X,Z) \to H_\lambda(X,Y)\bigr)}_{Q=\text{kernel}\bigl(H_\lambda(X,Y) \to H_{\lambda-1}(Y,Z)\bigr)} \to 1 $$ and so $\text{rank}(H_\lambda(X,Z)) = \text{rank}(K) + \text{rank}(Q)$.
Since $K$ is a quotient of $H_\lambda(Y,Z)$ we have $\text{rank}(K) \le \text{rank}(H_\lambda(Y,Z))$.
And since $Q$ is a subgroup of $H_\lambda(X,Y)$ we have $\text{rank}(Q) \le \text{rank}(H_\lambda(X,Y))$.
Putting this together, $$\text{rank}(H_\lambda(X,Z)) \le \text{rank}(H_\lambda(Y,Z)) + \text{rank}(H_\lambda(X,Y)) $$