Are ultralimits the Gromov-Hausdorff limits of a subsequence?

Let $(M_i,p_i)$ 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 $p \in X$ and subsequence of $(M_i,p_i)$ converging to $(X,p)$ in the pointed Gromov-Hausdorff sense?

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