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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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 PetruninCommented Aug 25, 2015 at 14:56
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$\begingroup$ The point $p$ will be just be the limit of the $p_i$. I think it's easier to consider the compact case. The non-compact case then follows by repeatedly passing to subsequences as you increase the radius of the balls. $\endgroup$– John HarveyCommented Aug 25, 2015 at 21:52
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