  We show that, for a certain large class of power-bounded $o$-minimal
$\mathcal{L}_T$-theories $T$ whose field of exponents is infinite-dimensional
as a vector space over the rationals, any definable set in a $T$-convex valued
field $(\mathcal{R}, \mathfrak{O})$ is in a precise sense the limit of a family
of $\mathcal{L}_T$-definable sets indexed over the residue field.
Alternatively, in the mainstream model-theoretic language, this says that if
$(\mathcal{R}', \mathfrak{O}')$ is an elementary substructure of $(\mathcal{R},
\mathfrak{O})$ and if the residue field of $\mathfrak{O}$ contains an element
that is infinitesimal relative to the residue field of $\mathfrak{O}'$ then any
set $A \subseteq (\mathcal{R}')^m$ definable in $(\mathcal{R}', \mathfrak{O}')$
is the trace of a set definable in $\mathcal{R}$.
