  One of the most powerful theorems in metric geometry is the Arzela-Ascoli
Theorem which provides a continuous limit for sequences of equicontinuous
functions between two compact spaces. This theorem has been extended by Gromov
and Grove-Petersen to sequences of functions with varying domains and ranges
where the domains and the ranges respectively converge in the Gromov-Hausdorff
sense to compact limit spaces. However such a powerful theorem does not hold
when the domains and ranges only converge in the intrinsic flat sense due to
the possible disappearance of points in the limit.
  In this paper two Arzela-Ascoli Theorems are proven for intrinsic flat
converging sequences of manifolds: one for uniformly Lipschitz functions with
fixed range whose domains are converging in the intrinsic flat sense, and one
for sequences of uniformly local isometries between spaces which are converging
in the intrinsic flat sense. A basic Bolzano-Weierstrass Theorem is proven for
sequences of points in such sequences of spaces. In addition it is proven that
when a sequence of manifolds has a precompact intrinsic flat limit then the
metric completion of the limit is the Gromov-Hausdorff limit of regions within
those manifolds. Applications and suggested applications of these results are
described in the final section of this paper.
