TWO chooses a countable set $A_n\subseteq G^*$ such that $\{x* U_{B_n}: x\in A_n\}$ is the set of distinct left cosets of $U_{B_n}$ in $G^*$. Since TWO has perfect information and for each $n$ $U_{B_{n+1}}\subseteq U_{B_n}$, TWO may select the sets $A_n$ such that for each $n$ we have $A_n\subseteq A_{n+1}$. We may assume for each $n$ that for each $x\in A_n$, if $x(\alpha)\neq {\sf id}_{G_{\alpha}}$ then $\alpha\in B_n$.

Now let $F$ be a winning perfect information strategy for TWO in the countable-1 game on $G^*$. We define a strategy $\sigma$ for TWO in ${{\sf G}_1}({{\mathcal{O}}_{nbd}},{\mathcal{O}})$ as follows:

Given ONE’s move ${\mathcal{O}}(U_{B_0})$ in ${{\sf G}_1}({{\mathcal{O}}_{nbd}},{\mathcal{O}})$ TWO first fixes $A_0$ as above, considered as a move of ONE of the countable-1 game. Then in that game TWO moves $x_0 = F(A_0)$. Then TWO responds in ${{\sf G}_1}({{\mathcal{O}}_{nbd}},{\mathcal{O}})$ with $\sigma({\mathcal{O}}(U_{B_0}) = x_0*U_{B_0}$.

In the next inning of ${{\sf G}_1}({{\mathcal{O}}_{nbd}},{\mathcal{O}})$ ONE moves ${\mathcal{O}}(U_{B_1})$. TWO first fixes the countble set $A_1$ as above and consider it as a move of ONE in the countable - 1 game on $G^*$. In that game TWO moves $x_1 = F(A_0,A_1)$. Then TWO responds in ${{\sf G}_1}({{\mathcal{O}}_{nbd}},{\mathcal{O}})$ with $\sigma({\mathcal{O}}(U_{B_0}),{\mathcal{O}}(U_{B_1})) = x_1*U_{B_1}$, and so on, as depicted in the following 