Question about $\aleph$-fixed point I am working through a proof on cardinals I found and can't reason some of the steps. 
The proposition is that there is an $\aleph$-fixed point, i.e. there is an ordinal $\alpha$ (which is necessarily a cardinal), so that $\aleph_{\alpha} = \alpha$. The proof goes as follows:
Let $\alpha_{0} = \aleph_{0}$ (or any other cardinal), $\alpha_{n + 1} = \aleph_{\alpha_{n}}$, and $\alpha = \sup \{ \alpha_{n} \mid n \in \omega \}$. Now if $\alpha = \alpha_{n}$ for some $n$, then $\alpha = \alpha_{n+1} = \aleph_{\alpha_{n}} = \aleph_{\alpha}$. Otherwise $\alpha$ is a limit ordinal and we have that $\aleph_{\alpha} = \sup \{ \aleph_{\xi} \mid \xi < \alpha\} = \sup\{ \aleph_{\alpha_{n}} \mid n \in \omega \} = \sup \{\alpha_{n + 1} \mid n \in \omega \} = \alpha$.
Now the limit case makes sense to me, but why on earth can we state that if $\alpha = \alpha_{n+1}$, then 
$$
\alpha = \alpha_{n+1} = \aleph_{\alpha_{n}} = \aleph_{\alpha}.
$$

I suppose the main issue I am having is why does $\alpha = \alpha_{n}$
  imply that $\alpha = \alpha_{n+1}$. 

After that, it is really just a matter of applying definitions.

I think the key piece is that the $\alpha_n$s are increasing: $\alpha_n\le\alpha_{n+1}$. 
More broadly, the following is true:

$\aleph_\beta\ge\beta$, for all $\beta$.

So we know $\alpha_n\le\alpha_{n+1}\le\alpha$ (the latter inequality since $\alpha$ is the sup of the $\alpha_n$s), so if $\alpha_n=\alpha$ then $\alpha_n=\alpha_{n+1}$ - and indeed $\alpha_n=\alpha_k$ for all $k\ge n$.
