Let $X$ be complete finite dimensional Alexandrov space with curvature bounded from below. Is it true that any two points can be connected by a shortest path? If this is not true in general, it it true under some assumptions on $X$? (E.g. this is certainly true for $X$ being a complete smooth Riemannian manifold.)
The answer is "yes", see e.g., p. 114 in [Plaut, Spaces of Wald-Berestovskii Curvature, http://link.springer.com.sci-hub.org/article/10.1007/BF02921569] - which is available online. It is claimed: "... If X is locally compact, Ascoli's theorem can be used to obtain the existence of minimal curves between all pairs of points ..."
Possibly, the other reference might be Y. Burago, M. Gromov, G. Perelman; Alexandrov's spaces with curvatures bounded from below I. Uspeki Mat. Nauk, 47 (1992), pp. 3–51. - which I did not find online