Suppose we have a set $B\subseteq 2^\omega\times\omega^\omega$ and a sequence $(x_n)$ in $2^\omega$ such that for each $n$, Player I (the one trying to get into the payoff set) has a winning strategy in the Gale-Stewart game where the payoff set is given by the corresponding slice: $B_{x_n}=\{f\in\omega^\omega:(x_n,f)\in B\}$.

**Question:** If $(x_n)$ converges to some $y$ in $2^\omega$, what can we say about the game with payoff set $B_y$? In particular, must Player I have a winning strategy in this game?

I am most interested in the cases when $B$ is *closed* or even *clopen*, in which case all of its slices are closed or clopen (and thus determined), respectively.

For what it's worth, I suspect the answer to my question is "no" in either case, but haven't been able to see why. Perhaps one could come up with an example where $B_y=\emptyset$?