  We study the interplay between algebraic and metric properties of
multi-parameter persistence modules (linear representations of the poset
$\mathbf{R}^n$ with $n \geq 2$). In the context of topological data analysis
and inference, a fundamental property of one-parameter persistence modules
(linear representations of $\mathbf{R}$) is that the operation of decomposing
them into their indecomposable summands is Lipschitz continuous, under suitable
choices of metrics on modules and on collections of indecomposables. This is
far from true of multi-parameter persistence modules: We show that any finitely
presentable multi-parameter persistence module can be approximated arbitrarily
well by an indecomposable module, and that modules that are sufficiently close
to an indecomposable decompose as a direct sum of an indecomposable and a
nearly trivial module. These results have several consequences of interest,
which provide strong motivation for approaching multi-parameter persistence in
a way that does not rely on directly decomposing modules by indecomposables.
