 Solution

Under the convention that a limit ordinal is nonzero and not a successor, the assertion is false at $\alpha =0$, since $\omega \cdot 0=0$. It is true for $\alpha >0$. If $\alpha =\eta +1$, then $\omega \alpha =\omega \eta +\omega$ has no greatest predecessor. If $\alpha$ is a nonzero limit, $\omega \alpha ={\sup }_{\eta <\alpha }\omega \eta$, again a nonzero limit. If zero is counted as a limit ordinal by convention, the unrestricted assertion is true with that convention.