Show that $\aleph_{\alpha}^{2}=\aleph_{\alpha}$ for all $\alpha$.

An infinite cardinal $m$ is called regular if it cannot be written as a sum of fewer than $m$ cardinals each of which is smaller than $m$. Prove that $\aleph_{0}$ and $\aleph_{1}$ are regular.

Is $\aleph_{2}$ regular? Is $\aleph_{\omega}$ regular? Justify your answers.