 Solution

True. Divide a limit ordinal $\lambda$ by $\omega$ using part a: $\lambda =\omega \alpha +n$ with $n<\omega$. If $n>0$, the right side is a successor, contrary to the limit assumption. Hence $n=0$ and $\lambda =\omega \alpha$. For a nonzero limit, $\alpha >0$; if zero is allowed as a limit, it is represented by $\alpha =0$.