  Given metric quotients $S$ and $S_n$, $n \in \mathbb{N}$, of a metric space
$X$, sufficient conditions are provided on the data defining them guaranteeing
that $S$ is the Gromov-Hausdorff limit of $S_n$. These conditions are
recognized within metric quotients of plane polygons determined by
side-pairings known as plain paper-folding schemes. In particular, concrete
examples are given of sequences of two-dimensional conic-flat spheres
converging to spheres that are conic-flat except around certain singularities,
some of them with unbounded curvature in the sense of comparative geometry.
