  The theory of Gromov-Hausdorff convergence is applied to sequences of
quotient rings of integers. It is shown the existence of limit rings (fields)
as the Gromov-Hausdorff limits of sequences of metric quotient rings. The
relation of these constructions with the field of the reals $\mathbb{R}$ is
discussed, showing that they are dense in $\mathbb{R}$ but that they cannot be
identified with the real field or with the rational field $\mathbb{Q}$, at
least when $\mathbb{R}$ and $\mathbb{Q}$ are endowed with the usual metric
structures. It is also shown that the limit rings can be endowed with an order
relation.
