Epsilon numbers: how to show $\varepsilon_{\omega_\alpha}=\omega_\alpha$ for $\alpha>0$? I would like to prove the equality (last property of Ordinals $\varepsilon$ numbers)
$$
\varepsilon_{\omega_\alpha} = \omega_\alpha,
$$
where as it is usual, $\varepsilon$ are the fixed-points of the exponential map
$$
f(\alpha)=(\omega_0)^\alpha .
$$
Here there is a related question about the cardinality of the class $\{\varepsilon_\alpha:\alpha<\omega_1\}$. For me the answer proves $\varepsilon_{\omega_1}\geq \omega_1$, but not the equality (just for me). Any way, I would like to know the general proof.
You are allow to use AC if it was necessary. 
I would like post my efforts, but I have no idea. If you explain me the case $\alpha=1$ maybe I could try to generalize the proof. for arbitary $\alpha$.
Thanks in advance
EDIT
Following Asaf Karagila's answer and this tips I have made some progress:
Proving $\mbox{card }\alpha\beta =\mbox{card }\alpha+\mbox{card }\beta$
is very easy, as he says, because $\aleph_\alpha+\aleph_\beta=\max\{\aleph_\alpha,\aleph_\beta\}$.
Now I need to prove $\mbox{card }\alpha^\beta = \mbox{card }\alpha\beta$.
I set $\alpha=\omega$ and I prove it for all $\beta$ by induction (later I'll use Cantor normal form to extend the result to all $\alpha$): $\omega^\omega$ is countable because is a countable union of countables.
$$\mbox{card }\omega^{\omega+1} = \mbox{card }\omega^\omega\omega = \mbox{card }\omega^\omega+\mbox{card }\omega $$
and the results follows again.
I suppose the result up to $\alpha$. If it is a succesor ordinal the proof follows the same computation than in the last case. So let $\alpha$ a limit ordinal. Then
$$
\mbox{card }\omega^\alpha = \mbox{card } \bigcup\{\omega^\xi:\xi<\alpha\} = \sum_{\xi<\alpha}\mbox{card }\omega^\xi = \sum_{\xi<\alpha}(\mbox{card }\omega + \mbox{card }\xi)
$$
and I don't know how to continue. In fact I'm not sure the second equality is correct.
Once I get this result I think I can extend it to all ordinals $\alpha^\beta$, since
$$
\omega^{\alpha'}\leq\alpha=\sum_{i=1}^n k_i\omega^{\alpha_i}<\omega^{\alpha'+1},
$$
where $\alpha'$ is de maximum of the $\alpha_i$'s. Because $\mbox{card }\omega^\alpha = \mbox{card }\omega^{\alpha+1}=\mbox{card }\alpha $
$$
\mbox{card }\alpha^\beta =\mbox{card } \omega^{(\alpha'\beta)}= \mbox{card }\alpha'\beta=\mbox{card }\alpha+\mbox{card }\beta .
$$
I don't know (yet) how to prove from this $\mbox{card }\varepsilon_\alpha=\mbox{card }\varepsilon_{\alpha+1}$ and why that implies $\mbox{card }\varepsilon_\alpha = \mbox{card } \alpha$.
 A: Prove by induction:

For all $\alpha$, $|\varepsilon_\alpha|=|\varepsilon_{\alpha+1}|$.

First note that for two infinite ordinals, $|\alpha\cdot\beta|=|\alpha|\cdot|\beta|$. Now by induction, we prove that $|\alpha^\beta|=|\alpha|\cdot|\beta|$. Suppose that for a fixed, infinite $\alpha$, $|\alpha^\beta|=|\alpha|\cdot|\beta|$ for all $\beta<\delta$.


*

*If $\delta$ is a limit, then $\alpha^\delta=\sup\{\alpha^\beta\mid\beta<\delta\}$, so $|\alpha^\delta|=\sup\{|\alpha^\beta|\mid\beta<\delta\}$, which is exactly $|\delta|\cdot\sup\{|\alpha^\beta|\mid\beta<\delta\}$, and since for $\beta<\delta$, we know that $|\alpha^\beta|=|\alpha|\cdot|\beta|$ we get the wanted result.

*If $\delta=\beta+1$, then $\alpha^\delta=\alpha^\beta\cdot\alpha$, and the result follows.
This, in turn, implies that $|\omega^\alpha|=|\alpha|$ for all $\alpha\geq\omega$. Now, $\varepsilon_{\alpha+1}=\sup\{\varepsilon_\alpha+1,\omega^{\varepsilon_\alpha+1},\ldots\}$, this is a countable supremum, and by the above we know that $\omega^{\varepsilon_\alpha+1}$ has the same cardinality as $\varepsilon_\alpha$, so by induction the whole sequence leading to $\varepsilon_{\alpha+1}$ also have the same cardinality, and therefore so does $\varepsilon_{\alpha+1}$.
Finally, this means that $|\varepsilon_\alpha|=|\alpha|+\aleph_0$. Again, we prove this by induction on $\alpha$. For $\alpha=0$, yes, this is well-known. For $\alpha=\beta+1$, this follows from the previous claim. Therefore for a limit ordinal, the only case we need to check is when $\alpha$ is itself a cardinal. But then we get that $\varepsilon_\alpha=\sup\{\varepsilon_\beta\mid\beta<\alpha\}$. This is a union of $\alpha$ many ordinals, and each has cardinality at most $\alpha$ (or strictly less, if $\alpha>\omega$). And again the conclusion follows.
This means that the $\omega_\alpha$th $\varepsilon$-number is the supremum of an increasing sequence of length $\omega_\alpha$, of ordinals which are all smaller than $\omega_\alpha$ itself. So it has to be $\omega_\alpha$. 
