  We consider sequences of compact Riemannian manifolds with uniform Sobolev
bounds on their metric tensors, and prove that their distance functions are
uniformly bounded in the H\"{o}lder sense. This is done by establishing a
general trace inequality on Riemannian manifolds which is an interesting result
on its own. We provide examples demonstrating how each of our hypotheses are
necessary. In the Appendix by the first author with Christina Sormani, we prove
that sequences of compact integral current spaces without boundary (including
Riemannian manifolds) that have uniform H\"{o}lder bounds on their distance
functions have subsequences converging in the Gromov--Hausdorff (GH) sense. If
in addition they have a uniform upper bound on mass (volume) then they converge
in the Sormani--Wenger Intrinsic Flat (SWIF) sense to a limit whose metric
completion is the GH limit. We provide an example of a sequence developing a
cusp demonstrating why the SWIF and GH limits may not agree.
