I'm trying to prove that the following statements are equivalent:
1.$\forall\alpha\in\mathbb{ON}\ \exists\ \kappa>\alpha$ is an inaccessible cardinal.
2.$\forall x\ \exists\ U\ x\in U$ and $U$ is a Grothendieck universe.
(1) $\rightarrow$ (2) is pretty straightforward. For the other direction, I was thinking to use induction on the ordinals; the successor case slides right by, but I have no ideas for the limit ordinals. Any thoughts are appreciated! Please let me know if definitions are required.