$\mathrm{cf}(\mathrm{cf}\alpha )=\mathrm{cf}\alpha$

Let

$\mathrm{cf}\alpha =\beta$ and be cofinal in $\alpha$, and

$\mathrm{cf}\beta =\gamma$ and be cofinal in $\beta$.

The claim of the theorem $\mathrm{cf}(\mathrm{cf}\alpha )=\mathrm{cf}\alpha$ means that $\gamma =\beta$; we prove this fact.

Suppose $\gamma \ne \beta$. Then by $\mathrm{cf}\delta \le \delta$.

Now, is seen to be confinal in $\alpha$, which means that , a contradiction. Therefore, $\gamma =\beta$.