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We consider sequences of metrics, g j , on a compact Riemannian manifold, M, which converge smoothly on compact sets away from a singular set S ⊂ M, to a metric, g ∞ , on M \ S . We prove theorems which describe when M j = (M, g j ) converge in the Gromov-Hausdorff sense to the metric completion, (M ∞ , d ∞ ), of (M \ S , g ∞ ). To obtain these theorems, we study the intrinsic flat limits of the sequences. A new method, we call hemispherical embedding, is applied to obtain explicit estimates ondoi:10.4310/cag.2013.v21.n1.a2 fatcat:bkory64snvcp3clzwqy7oqomwm