  The algebraic stability theorem for $\mathbb{R}$-persistence modules is a
fundamental result in topological data analysis. We present a stability theorem
for $n$-dimensional rectangle decomposable persistence modules up to a constant
$(2n-1)$ that is a generalization of the algebraic stability theorem, and also
has connections to the complexity of calculating the interleaving distance. The
proof given reduces to a new proof of the algebraic stability theorem with
$n=1$. We give an example to show that the bound cannot be improved for $n=2$.
We apply the same technique to prove stability results for zigzag modules and
Reeb graphs, reducing the previously known bounds to a constant that cannot be
improved, settling these questions.
