Existence of a fixed point in the map Two maps of the same region drawn to different scales are superimposed so that the smaller map lies entirely inside the larger. Prove that there is precisely one point on the small map that lies directly over a point on the large map that represents the same place of the region.

The larger map can be thought of as a domain D in the plane. The change of scale from one map to another is a contraction, and since the smaller map is placed inside the larger, the contraction maps D to D. A point such as the one described in the statement is a fixed point for this contraction.

Fixed point theorem: Let X be a closed subset of $R^n$ (or in general of a complete metric space) and f : X → X a function with the property that f (x) − f (y) ≤ cx − y for any x, y ∈ X, where 0 < c < 1 is a constant. Then f has a unique fixed point in X,

By the above theorem, the point exists and is unique.