= Solution
Under the convention that a <limit ordinal> is nonzero and not a successor, the assertion is \b[false at $\alpha=0$], since $\omega\cdot0=0$. 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.
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