Gale's 1979 proof that Hex's no-draw property implies the Brouwer fixed-point theorem uses a covering argument on an arbitrarily fine Hex board

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In Section 3 of "The Game of Hex and the Brouwer Fixed-Point Theorem" (The American Mathematical Monthly, 1979), David Gale opens the first half of his equivalence proof — "We first show, that 'Hex' implies 'Brouwer.'" — with an approximation argument rather than a proof by contradiction. Given a continuous $f: I^2 \to I^2$, compactness of the square reduces the theorem to producing points that move arbitrarily little: "From compactness of $I^2$ it suffices to show that for any $\varepsilon > 0$ there exists $x \in I^2$ such that $|f(x) - x| < \varepsilon$." Uniform continuity supplies a $\delta < \varepsilon$, and Gale then takes a Hex board $B_k$ fine enough that $1/k < \delta$ — this is where "arbitrarily fine" enters, as a choice made per $\varepsilon$, not a limit. The board's vertices are sorted into four sets $H^+, H^-, V^+, V^-$ by the direction $f$ moves them, not by who owns them: "a vertex $z$ belongs to $H^+, H^-, V^+, V^-$ according as $z/k$ is moved by $f$ at least $\varepsilon$ units to the right, left, up, or down." Gale shows $H^+$ and $H^-$ are not contiguous, and likewise $V^+$ and $V^-$, so no connected subset of $H = H^+ \cup H^-$ meets both the E and W boundaries and none of $V = V^+ \cup V^-$ meets both N and S. The Hex Theorem (Hex cannot end in a draw) then forbids $H$ and $V$ from covering the board at all — and any vertex they miss is a point moved less than $\varepsilon$. The proof closes on: "By the Hex Theorem, therefore, the sets H and V do not cover Bk, completing the proof."

The no-fixed-point assumption belongs to the converse direction, not this one: it is in the Brouwer-implies-Hex half that Gale argues by contradiction, assuming no winning path exists and building from it a simplicial map with no fixed point.

This is one of two proof directions behind claim-gale-1979-hex-draw-impossibility-equivalent-to-brouwer-fixed-point's general equivalence claim. It is also the half that carries Gale's stated purpose for the paper: "Our main purpose is to show that a classical result of topology, the celebrated Brouwer Fixed-Point Theorem, is an easy consequence of the fact that Hex... cannot end in a draw." Gale frames this specific direction as his own contribution — "The proof going in the other direction only occurred to me recently" — in contrast to the converse direction, which he attributes to a suggestion from John Stallings, modified by Michael Todd (see claim-gale-1979-brouwer-implies-hex-credited-to-stallings-todd). The result sits alongside the vault's other Hex-solving methods: Shannon and Moore's analog machine (claim-shannon-moore-1950-analog-hex-machine-move-as-saddle-point) and Nash's non-constructive existence proof (claim-nash-hex-first-player-win-proof-is-non-constructive).