I continue reading that book on exercises from set theory: Here on the page 112 they define a Countryman type. On the page 439 they go on to prove that every Countryman is Specker. I have a problem to understand some parts of the proof: first, why there must be $\alpha < \alpha'$ with $i(\alpha)=i(\alpha')$ ? Second why such $\beta$ and $\beta'$ can be found ?
The existence of such $\alpha$ and $\alpha'$ is just Pigeonhole again. There are $\aleph_1$ different values of $\alpha$ but only $\aleph_0$ possible values of $i(\alpha)$ so there must exist distinct $\alpha$ and $\alpha'$ with $i(\alpha)=i(\alpha')$, and you may assume without loss of generality that $\alpha<\alpha'$.
Since $i(\alpha')=i$, there are uncountably many different $\beta'$ such that $\langle \alpha',\beta'\rangle\in A_i$, so just pick one of them. Then since $i(\alpha)=i$, there are also uncountably many $\beta$ such that $\langle \alpha,\beta\rangle\in A_i$. In particular, there is such a $\beta$ which is greater than $\beta'$, since there are only countably many ordinals less than $\beta'$.