I've got stuck trying to prove this result about cofinality of limit ordinals. I am trying to prove:
For $Lim(\beta)$ and $\alpha>0$, $$cf(\alpha * \beta) = cf(\alpha + \beta) = cf(\beta)$$
I tried to prove this by induction on $\alpha$ and I'm not sure if this is the right way to go about it. What I've done so far is:
Base case: $\alpha = 1$
Clearly we have $cf(\alpha * \beta) = cf(\beta)$ and also as $Lim(\beta)$ we have $cf(1 + \beta) = cf(\beta)$
Assume as an inductive hypothesis that the statement is true for all $\gamma < \alpha$. Then
$$cf(\alpha + \beta) = cf((\gamma + 1) + \beta) = cf(\gamma + (1 + \beta)) = cf(\gamma + \beta) = cf(\beta)$$
I think the third equivalence holds as $\beta$ is a limit ordinal and then the fourth holds from the inductive hypothesis.
I tried something similar for $cf(\alpha*\beta)$ but got stuck:
$$cf(\alpha*\beta) = cf((\gamma + 1)*\beta) = cf(\gamma*\beta + \beta)$$
and then I wasn't sure where to go from there.
It would be great to know if I've been going down the right path and how to finish the proof, or if I've been going completely wrong and how I can prove this result. Thanks a lot!
Consider the map $\phi: \beta \rightarrow \alpha * \beta, \space \phi(\gamma) = \alpha *\gamma$.
$\phi$ is strictly increasing, hence $\beta$ is order-isomorphic to $\phi(\beta)$, which is cofinal in $\alpha * \beta$.
Hence $\text{cf}(\beta) = \text{cf}(\phi(\beta)) = \text{cf}(\alpha *\beta)$.
Similar for $\text{cf}(\beta) = \text{cf}(\alpha + \beta)$