There is a completely elementary and very elegant proof of the Brower fixed point theorem based on a beautiful combinatorial result called Sperner lemma. For details I recommend Section 2.3, page 72 of the beautiful book

V.V. Prasolov: Elements of Combinatorial and Differential Topology, Graduate Studies in Mathematics, vol. 74, Amer. Math. Soc., 2006

The proof is constructive and it leads to an algorith for producing a sequence of points converging to a fixed point of the map. Prasolov attributes this approach to

B. Kuratowski, C. Knaster, C. Mazurkiewicz: Ein Beweis des Fixpunktsatz fur n_dimensionale Simplexe, Funt. Math. 14(1929), 132-137.

Note This does not really answer your question.

There is a completely elementary and very elegant proof of the Brower fixed point theorem based on a beautiful combinatorial result called Sperner lemma. For details I recommend Section 2.3, page 72 of the beautiful book

V.V. Prasolov: Elements of Combinatorial and Differential Topology, Graduate Studies in Mathematics, vol. 74, Amer. Math. Soc., 2006

The proof is constructive and it leads to an algorithm for generating a sequence of points converging to a fixed point of the map. Prasolov attributes this approach to

B. Kuratowski, C. Knaster, C. Mazurkiewicz: Ein Beweis des Fixpunktsatz fur n-dimensionale Simplexe, Funt. Math. 14(1929), 132-137.

Note This does not really answer your question.

A proof along the lines you highlighted can be fond in J. Milnor's book, Topology from a differentiable viewpoint.

There is a completely elementary and very elegant proof of the Brower fixed point theorem based on a beautiful combinatorial result called Sperner lemma. For details I recommend Section 2.3, page 72 of the beautiful book

V.V. Prasolov: Elements of Combinatorial and Differential Topology, Graduate Studies in Mathematics, vol. 74, Amer. Math. Soc., 2006

The proof is constructive and it leads to an algorith for producing a sequence of points converging to a fixed point of the map. Prasolov attributes this approach to

B. Kuratowski, C. Knaster, C. Mazurkiewicz: Ein Beweis des Fixpunktsatz fur n_dimensionale Simplexe, Funt. Math. 14(1929), 132-137.

There is a completely elementary and very elegant proof of the Brower fixed point theorem based on a beautiful combinatorial result called Sperner lemma. For details I recommend Section 2.3, page 72 of the beautiful book

V.V. Prasolov: Elements of Combinatorial and Differential Topology, Graduate Studies in Mathematics, vol. 74, Amer. Math. Soc., 2006

The proof is constructive and it leads to an algorith for producing a sequence of points converging to a fixed point of the map. Prasolov attributes this approach to

B. Kuratowski, C. Knaster, C. Mazurkiewicz: Ein Beweis des Fixpunktsatz fur n_dimensionale Simplexe, Funt. Math. 14(1929), 132-137.

Note This does not really answer your question.