Prove Sperner’s lemma in dimension \(1\), namely on a line.
The simplex in this case is just a segment, the triangulation is
subdivision of the segment into multiple small segments, and the
conditions of a Sperner’s coloring are the following:
There are only two colors;
The opposite ends of the main segment are colored differently;
Then one needs to prove that there exists a small segment with two ends colored in different colors. In particular there is an odd number of such small segments.