I'm trying to prove Exercise 3.5 in Set Theory by Thomas Jech: $$ \Gamma(\alpha\times\alpha) \leq \omega^\alpha $$ where $\Gamma(\alpha\times\beta)=\operatorname{otp}(\{(\zeta,\delta)\mid(\zeta,\delta)\lt(\alpha\times\beta)\})$ and $$(\zeta,\delta)\lt(\alpha,\beta) \iff \begin{cases} \max(\zeta,\delta)\lt \max(\alpha,\beta)\\ \max(\zeta,\delta)= \max(\alpha,\beta)\text{ and } \zeta\lt\alpha\\ \max(\zeta,\delta)= \max(\alpha,\beta)\text{ and } \zeta=\alpha \text{ and }\delta\lt\beta \end{cases}$$ Here's what I've tried so far:
- If $\alpha=0$ , then $\Gamma(\alpha\times\alpha) = 0\le1=\omega^0$.
- If $\alpha>0$ is finite, then $\Gamma(\alpha\times\alpha)$ is also finite, hence $\Gamma(\alpha\times\alpha) \lt \omega \le\omega^\alpha$.
Then I'm not too sure where to go from here