Prove the limit step for $\Gamma (\alpha, \alpha)\le \omega^\alpha$

Question

I'm working through Jechs Set Theory right now and encountered an exercise (3.5) to prove that the order type of $\Gamma(\alpha, \alpha):=\{(\xi, \eta): (\xi, \eta)<(\alpha, \alpha)\}$ is less than or equal to $\omega^\alpha$ for any ordinal $\alpha$. The $<$ relation for $(\xi, \eta)$ is defined as: $$(\alpha, \beta)<(\gamma, \delta) \Leftrightarrow (\max \{\alpha, \beta\}<\max\{\gamma, \delta\})\lor(\max \{\alpha, \beta\}=\max\{\gamma, \delta\}\land \alpha < \gamma)\lor(\max \{\alpha, \beta\}=\max\{\gamma, \delta\}\land \alpha =\gamma \land \beta<\delta)$$ I tried to prove it using transfinite induction, but I wasn't able to do the limit step. (Though it seems to be the correct approach considering this question concerning the same exercise: Show that $\Gamma(\alpha\times\alpha)\leq\omega^{\alpha}$. )