If $X$ is an Alexandrov space of curvature bounded below by a real number $k$, is it true that any geodesic in the tangent cone $T\_pX$ can be realized as a limit of geodesics when we view $T\_pX$ as the Gromov-Hausdorff limit of rescalings of neighborhoods of $p$?


More generally, if $X\_i$ is a sequence of Alexandrov spaces with curvatures bounded below uniformly by $k$ and $\dim X\_i=n$, such that $X\_i$ converges in the Gromov-Hausdorff sense to an Alexandrov space $X$ with $\dim X=n$ (i.e., the sequence is non-collapsing), is it true that any geodesic in $X$ is the limit of a sequence of geodesics in $X\_i$?



Is any geodesic in the tangent cone of an Alexandrov space a limit geodesic?
The answer is "yes" assuming you are interested in *minimizing geodesics*.
Moreover the statement holds in the collapsing case as well.


Fix two points $p$ and $q$ in the limit space $X$.
If there is unique geodesic $[p,q]$ connecting these two points then we can choose arbitrary converging sequences $X\_n\ni p\_n\to p$ and $X\_n\ni q\_n\to q$ --- any sequence of geodesics $\gamma\_n=[p\_n,q\_n]$ does the trick.


If there are many geodesics, fix $p',q'\in [p,q]$ close to the ends.
By comparison, the geodesic $[p',q']$ is uniquely defined. 
Therefore by the argument above $[p',q']$ is a limit of geodesics in $X\_n$.


Repeat this argument for a sequence of pairs of points $p'\_k\to p$ and $q'\_k\to q$.
You get double sequence of geodesics $\gamma\_{n,k}$ in $X\_n$.
Applying diagonal procedure you get an approximation of $[p,q]$ by a sequence $\gamma\_{n,k(n)}$.


