Bounded-orbit fixed point for a strictly distance-decreasing Euclidean map
= Bounded-orbit fixed point for a strictly distance-decreasing Euclidean map
For a strictly distance-decreasing map of a finite-dimensional <Euclidean space>, one bounded forward <orbit> suffices for existence of a <fixed point>. Its closure is compact by the <Heine-Borel theorem> and invariant by continuity, so <strict distance decrease on a compact metric space> applies. Strict distance decrease gives uniqueness throughout the whole space, not merely the orbit closure.