Cofinality of cardinals

Starting out with $\aleph_0$, since the union of finitely many finite sets is finite, it follows that $\text{cof}(\aleph_0)=\aleph_0$.

For successor cardinals, since $\aleph_{\alpha+1}$ means $(\aleph_\alpha)^+$, which is a regular cardinal (this uses AC), we know $\text{cof}(\aleph_{\alpha+1})=\aleph_{\alpha+1}$.

The remaining case is limit cardinals, or $\aleph_\lambda$ for a limit ordinal $\lambda$. In this case, $\aleph_\lambda=\text{sup}_{\alpha\lt\lambda}\aleph_\alpha$ is the limit of a $\lambda$-sequence of smaller cardinals, and so by passing to the shortest possible subsequence of this sequence, we see that $\text{cof}(\aleph_\lambda)=\text{cof}(\lambda)$. So for example, $\text{cof}(\aleph_{\aleph_3+\omega^3})=\omega$, since there is an $\omega$-sequence unbounded in the ordinal $\aleph_3+\omega^3$.