If $\aleph_\alpha=\alpha$, then $\alpha$ is a limit ordinal

Question

If $\aleph_\alpha=\alpha$, then $\alpha$ is a limit ordinal.

My attempt:

Assume the contrary that $\alpha$ is not a limit ordinal. Then $\alpha$ is a successor ordinal and thus $\alpha=\beta+1$ for a unique $\beta$.

It follows that $\beta \bigcup\{\beta\}=\aleph_\alpha$. Thus $|\aleph_\alpha|=|\beta|$ or $\aleph_\alpha=|\beta|$. This means $\aleph_\alpha$ is equipotent to a smaller ordrinal. This is clearly a contradiction.

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My proof is quite short and I wonder if it contains any logical flaw/gap. Thank you for your help!

Answer 1

Yes your argument follows. In general, any infinite cardinal is a limit ordinal.