Result: On the Set of Points Represented by Harmonic Subseries

We help Alice play a certain "convergence game" against Bob and win the prize, which is a constructive solution to a problem by Erdős and Graham, posed in their 1980 book on open questions in combinatorial number theory. Namely, after several reductions using peculiar arithmetic identities, the game outcome shows that the set of points \[ \Big(\sum_{n\in A}\frac{1}{n}, \sum_{n\in A}\frac{1}{n+1}, \sum_{n\in A}\frac{1}{n+2}\Big), \] obtained as $A$ ranges over infinite sets of positive integers, has a non-empty interior. This generalizes a two-dimensional result by Erdős and Straus.
14 pages; v2: the proof is rewritten as a strategic two-player game; the exposition is less formal and (hopefully) more entertaining; an explicit ball in the interior is constructed; Mathematica notebook that supports computation is updated; v3: minor changes following referees' suggestions