Show $\alpha$ is a limit ordinal $\leftrightarrow \alpha \neq 0$ and $\cup \alpha = \alpha$

Question

$\alpha$ is a limit ordinal $\leftrightarrow \alpha \neq 0$ and $\cup \alpha = \alpha$

Sorry if this question has been asked already but I couldn't find it on this site.

I assume by definition of a limit ordinal you can state it does not equal $0$. As for showing $\cup \alpha = \alpha$, you have the fact that Sup($\alpha$) = $\cup \alpha$ but I don't really know where to go from there, especially if I want a concrete rigorous proof.

Thanks in advance.

Answer 1

HINT: If $\alpha=\beta+1$, show that $\bigcup\alpha=\beta$.