Are ultralimits the Gromov-Hausdorff limits of a subsequence?

Let (Mi,pi) be a sequence of n-dimensional Riemannian manifolds with lower Ricci curvature bound 1. Fix a non-orincipal ultrafilter and let X be the ultralimit of the sequence.
Does there exists a pX and subsequence of (Mi,pi) converging to (X,p) in the pointed Gromov-Hausdorff sense?

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Source : Link , Question Author : dg.jan , Answer Author : Community

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