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 G is a non-empty open set. It is easy to show that G must contain some dyadic interval [n/2^{k},(n+1)/2^{k}], for integer n and (large enough) k. By playing the first k digits of n/2^{k}, Alice guarantees a win -- any digits added will stay within the interval, so x will be contained in G.