the root of the difficulty is explained in as follows. it is relatively simple to prove that the jordan curve theorem holds for every jordan polygon (lemma 1), and every jordan curve can be approximated arbitrarily well by a jordan polygon (lemma 2). a jordan polygon is a polygonal chain, the boundary of a bounded connected open set, call it the open polygon, and its closure, the closed polygon. consider the diameter formula_2 of the largest disk contained in the closed polygon. evidently, formula_2 is positive. using a sequence of jordan polygons (that converge to the given jordan curve) we have a sequence formula_4 "presumably" converging to a positive number, the diameter formula_2 of the largest disk contained in the closed region bounded by the jordan curve. however, we have to "prove" that the sequence formula_4 does not converge to zero, using only the given jordan curve, not the region "presumably" bounded by the curve. this is the point of tverberg's lemma 3. roughly, the closed polygons should not thin to zero everywhere. moreover, they should not thin to zero somewhere, which is the point of tverberg's lemma 4.