Ordinal arithmetic $\alpha = \omega + \alpha$

Question

I am trying to solve the ordinal equation $\alpha = \omega + \alpha$ (with indeterminate $\alpha$). I want to show that $\alpha = \omega + \alpha$ iff $ \alpha \geq {\omega}^2$. The right-to-left implication is pretty easy but I am stuck on the left-to-right implication. I have tried contraposition to show that $\alpha < \omega + \alpha$ but I don't see the argument that would work (the only trivial case is when $\alpha$ is finite of course).