In this paper, we prove that there is no branch point in the Lorentz space (M, d) which is the limit space of a sequence {($M_{alpha},d_{alpha}$)} of compact globally hyperbolic interpolating spacetimes with $C^{pm}_{alpha}$-properties and curvature bounded below. Using this, we also obtain that every maximal timelike geodesic in the limit space (M, d) can be expressed as the limit curve of a sequence of maximal timelike geodesics in {($M_{alpha},d_{alpha}$)}. Finally, we show that the limit space (M, d) satisfies a timelike triangle comparison property which is analogous to the case of Alexandrov curvature bounds in length spaces.