  We investigate, in ZFC, the behavior of abstract elementary classes (AECs)
categorical in many successive small cardinals. We prove for example that a
universal $\mathbb{L}_{\omega_1, \omega}$ sentence categorical on an end
segment of cardinals below $\beth_\omega$ must be categorical also everywhere
above $\beth_\omega$. This is done without any additional model-theoretic
hypotheses (such as amalgamation or arbitrarily large models) and generalizes
to the much broader framework of tame AECs with weak amalgamation and coherent
sequences.
