  We introduce an algorithm which, given probabilities $\mu \leq_{\text{cx}}
\nu$ in convex order and defined on a separable Banach space $B$, constructs
finitely-supported approximations $\mu_n \to \mu, \nu_n\to \nu$ which are in
convex order $\mu_n \leq_{\text{cx}} \nu_n$. We provide upper-bounds for the
speed of convergence, in terms of the Wasserstein distance. We discuss the
(dis)advantages of our algorithm and its link with the discretisation of the
Martingale Optimal Transport problem, and we illustrate its implementation with
numerical examples. We study the operation which, given $\mu$/$\nu$ and some
(finite) partition of $B$, outputs $\mu_n$/$\nu_n$, showing that applied to a
probability $\gamma$ and to all partitions it outputs the set of all
probabilities $\zeta \leq_{\text{cx}} \gamma$.
