I have been trying to prove that if $k$ is an infinite cardinal number, then its successor $k^+$ is regular, that is $\operatorname{cf}(k^+)=k^+$.
In my class we have defined cofinality in the following way:
$$\begin{align*} \operatorname{cf}(k)&=\min\left\{I \subseteq k\mid \text{for some family of cardinal numbers } (k_i)_{i \in I}\\ (\forall i \in I) [k_i<k]\,\&\,\sum_{i\in I} k_i = k\right\} \end{align*}$$
I tried a proof by contradiction, supposing that $\operatorname{cf}(k^+)<k^+$ and using König's theorem, all to no avail. The problem can be found in the book Notes on Set Theory by Yannis Moschovakis as x.9.17.