Basic Analysis I — Real Analysis · Problem 7.6.6

The key insight is that Banach's fixed-point theorem requires both conditions: a contraction (strict shrinking) and completeness. For part (a), you need a space where the iterates of a contracting map converge to a limit that lies outside the space itself—think of a map on an open interval that pushes points toward a boundary they never reach. For part (b), a 1-Lipschitz map can slide points along without ever fixing them if the space has enough "room to escape"—consider a translation or a map that shifts every point by a constant distance in a complete space like $\mathbb{R}$.

The key insight is that Banach's fixed-point theorem requires both conditions: a contraction (strict shrinking) and completeness. For part (a), you need a space where the iterates of a contracting map converge to a limit that lies outside the space itself—think of a map on an open interval that pushes points toward a boundary they never reach. For part (b), a 1-Lipschitz map can slide points along without ever fixing them if the space has enough "room to escape"—consider a translation or a map that shifts every point by a constant distance in a complete space like $\mathbb{R}$.