>>12631178This is a contractive mapping on , where , which is a complete metric space. By the Banach Fixed Point Theorem, the sequence defined by indefinitely recursing the contraction mapping, starting at any initial point, must converge to the (existing and unique) fixed point of the contractive mapping. Because can be made indefinitely large, and because that interval includes the fixed points of any and all intervals produced with smaller and because the fixed point must be unique for each interval, we know that only one fixed point for the contraction must exist on , and that the recursive sequence must converge to it from any initial point.
By simple algebra, 2 must be this stable fixed point, so half of it must be 1.
