Let $(M_i,p_i)$ be a sequence of $n$dimensional Riemannian manifolds with lower Ricci curvature bound $1$. Fix a nonorincipal 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 GromovHausdorff sense?
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3$\begingroup$ Yes it is true and follow directly from the definition. However it is not true that the sequence can be found in the ultrafilter, see mathoverflow.net/questions/111842 $\endgroup$ – Anton Petrunin Aug 25 '15 at 14:56

$\begingroup$ The point $p$ will be just be the limit of the $p_i$. I think it's easier to consider the compact case. The noncompact case then follows by repeatedly passing to subsequences as you increase the radius of the balls. $\endgroup$ – John Harvey Aug 25 '15 at 21:52