  We consider collections of Lagrangian submanifolds of a given symplectic
manifold which respect uniform bounds of curvature type coming from an
auxiliary Riemannian metric. We prove that, for a large class of metrics on
these collections, convergence to an embedded Lagrangian submanifold implies
convergence to it in the Hausdorff metric. This class of metrics includes
well-known metrics such as the Lagrangian Hofer metric, the spectral norm and
the shadow metrics introduced by Biran, Cornea and Shelukhin arXiv:1806.06630.
The proof relies on a version of the monotonicity lemma, applied on a
carefully-chosen metric ball.
