Geodesic completeness, cosmological bounces and inflation

The question of geodesic completeness of cosmological spacetimes has recently received renewed scrutiny. A particularly interesting result is the observation that the well-known Borde-Guth-Vilenkin (BGV) theorem may misdiagnose geodesically complete cosmologies. We propose a simple amendment to the BGV theorem which addresses such loopholes while retaining much of its generality. We give straightforward proofs of some recently offered conjectures concerning (generalized) Friedmann-Lema\^itre-Robertson-Walker spacetimes: geodesic completeness implies (i) the existence of a bounce, loitering phase or an emergent cosmology, and (ii) a phase of accelerated expansion with strictly increasing Hubble rate. Our results are purely kinematic and do not assume general relativity or energy conditions.