The mountain climbing lemma is a useful tool, that is at the same time tightly related with inscribed figures in closed curves. It is about two mountaineers, both standing on different sides of a mountain range. They want to cross it, but their elevation must remain equal at all times. If we replace the mountain range with a function f, we shall show in the first part of this work that if f does not change sign and consists of a finite number of monotone non-increasing or non-decreasing pieces, then the mountain-climbing lemma holds and the mountaineers can indeed cross it in the above explained way. In the second part we will see the geometric usage of this lemma. We will introduce the problem of inscribing a square into a closed curve, also known as the square peg problem. It exists for over a hundred years and still has not been solved in a completely general form. With the help of the mountain climbing lemma we will find inscribed rhombi in polygons, but along the way we will discover many surprising facts about inscribed triangles and rectangles in mostly planar polygons.
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