Trying to show that the von Neumann universes $R_{\omega+ \alpha}$ have cardinality $\beth_\alpha$.
Here is my attempt:
Proof by induction on $\alpha$. For $\alpha = 0$, $|R_{\omega+ 0}| = |R_{\omega}| = \omega = \beth_0$.
For $\alpha = \beta+1$, $|R_{\omega+ \alpha}| = |\mathcal{P}(R_{\omega+\beta})| = {2}^{|R_{\omega+\beta}|}$. By the IH, we have ${2}^{|R_{\omega+\beta}|} = {2}^{ \beth_\beta} = \beth_{\alpha}$. Hence, $|R_{\omega+ \alpha}| = \beth_{\alpha}$.
For $\alpha = \bigcup\limits_{\beta < \alpha} \beta$, $\beth_\alpha = \sup\limits_{\beta < \alpha}\{ \beth_\beta \}$. By the IH, $|R_{\omega+\beta}| = \beth_\beta$ for all $\beta < \alpha$. So, $\sup\limits_{\beta < \alpha}\{ |R_{\omega+\beta}| \} = \sup\limits_{ \beta < \alpha}\{ \beth_\beta \}$.
I'm not sure how to finish the limit case though. Any help would be great.