Metric extraction of a midpoint by shrinking diameters
= Metric extraction of a midpoint by shrinking diameters
For $v,w$ in a real normed space, start with the points at distance $\|v-w\|/2$ from both endpoints and repeatedly retain those within half the previous <diameter of a metric space>[diameter] of every retained point. Reflection about $(v+w)/2$ preserves every stage, the midpoint belongs to every stage, and the diameters decrease by a factor at least two. Their intersection is therefore exactly the midpoint.