This node is part of the game proof of the Baire category theorem, and won't make any sense unless you're coming from there!

Suppose Bob is guaranteed a win when playing on any one of G_{1}, G_{2}, ... Let G=G_{1}∪G_{2}∪... be their union; we wish to show how Bob is guaranteed a win even when playing on all of G (recall that "larger" target sets G should help Alice, not Bob). To that end, we outline Bob's winning strategy for G.

On the first move, Alice has already played some bit sequence A_{1}. At the very least, Bob must play to ensure that x won't be in G_{1}, so Bob pretends he's just playing G_{1}, and makes the appropriate response B_{1,1} to Alice's A_{1} when playing G_{1}.

Alice responds with some sequence A_{2}, and the position looks like `0. A`_{1} B_{1,1} A_{2}. So Bob first works out the appropriate response B_{2,1} to `0.A`_{1} B_{1,1} A_{2} when playing G_{1}, and then (since he doesn't want x to be in G_{2}, either) the appropriate response B_{2,2} to `0.A`_{1} B_{1,1} A_{2} B_{2,1} when playing G_{2} (note that this could have been Alice's first move when playing G_{2}, so Bob's winning strategy has an appropriate response to it!). Bob then plays `B`_{2,1} B_{2,2}.

Continuing, after Alice's n+1'th move, the position is

`0. A`_{1} B_{1,1} A_{2} B_{2,1} B_{2,2} A_{3} ... A_{n} B_{n,1} B_{n,2} ... B_{n,n} A_{n+1},

and Bob must play. Alice's n+1'th move when playing G

_{1} could have been

`B`_{n,2} ... B_{n,n} A_{n+1}, so Bob's winning strategy for G

_{1} tells him to play some move B

_{n+1,1}; now Bob pretends Alice's n'th move when playing G

_{2} was

`B`_{n,3} ... B_{n,n} A_{n+1} B_{n+1,1}, and his winning strategy for G

_{2} tells him to play some B

_{n+1,2}; Bob continues in this manner, letting B

_{n+1,k} be the response his winning strategy for G

_{k} tells him to play if Alice's move n+2-k when playing Gk had been

`B`_{n,k+1} ... B_{n,n} A_{n+1} B_{n+1,1} ... B_{n+1,k-1}. After getting the n+1 bit strings B

_{n+1,1}, ..., B

_{n+1,n+1}, Bob plays them all in sequence (still a finite digit string, so it's a legal move!).

Now consider the x that results from Bob's strategy. It cannot be in any G_{k}, since Bob has been following a line of play guaranteed (assuming Alice plays in a certain way, which includes some of Bob's responses) to keep x out of G_{k}. So x cannot be in any of G_{k}, hence Bob has a guaranteed win when playing the union of the G_{k}'s.