Explain that there is exactly one point on the map that is vertically above the point it depicts You are standing with a map in your hand inside the area depicted on the map. Explain that there is exactly one point on the map that is vertically above the point it depicts.

I am familiar with Banach's Fixed Point Theorem or Contraction Mapping Theorem. But I dont know how solve this problem. How do I show that there is such point? Other than to draw
 A: Denote by $X\subset {\mathbb R}^2$ the area that is depicted by the topographic "map" $R$. This "map" itself is a rectangle  that is "laid over" a part of  $X$. We then have two mathematical maps, namely (i) the cartographic map (realized by the cartographers and the printer)
$$\phi_1: \quad X\to R\ ,$$ which is a contraction by a factor ${1\over 1000}$, say, and (ii) the projection $$\phi_2:\quad R\to X$$ of $R$ to the ground on which we have laid the "map" $R$. Given a point $p\in R$ the image point $\phi_2(p)$ can be physically determined by punching a needle at $p$ through the paper down to $X$. It follows that $\phi_2$ is an isometry.
The composition
$$f:\quad X\to X, \qquad x\mapsto \phi_2\bigl(\phi_1(x)\bigr)$$
is a contraction of the complete metric space $X$ with Lipschitz constant ${1\over1000}$. By Banach's fixed point theorem there is exactly one point $\xi\in X$ with $$\phi_2\bigl(\phi_1(\xi)\bigr)=f(\xi)=\xi\ .$$
But this is saying that punching a needle through the point $\phi_1(\xi)\in R$ hits exactly the point $\xi\in X$.
By the way: There is a (quite deformed) homunculus in your brain that is a mental picture of your bodily self. Exactly one spot of this homunculus depicts the very spot where it is located.
