  In this paper, we consider a fixed metric space (possibly an oriented
Riemannian manifold with boundary) with an increasing sequence of distance
functions and a uniform upper bound on diameter. When the metric space endowed
with the pointwise limit of these distances is compact, then there is uniform
and Gromov-Hausdorff (GH) convergence to this limit. When the metric space also
has an integral current structure of uniformly bounded total mass (as is true
for an oriented Riemannian manifold with boundary that has a uniform bound on
total volume), we prove volume preserving intrinsic flat convergence to a
subset of the GH limit whose closure is the whole GH limit. We provide a review
of all notions and have a list of open questions at the end. Dedicated to
Xiaochun Rong.
