Sometimes, it's fun to muse about basic stuff. I was in class and suddenly thought about how to prove the statement in the title.

Consider 4 points on the plane not coplanar. Choose 3, and construct the circumcircle. This will be a circle lying on our sphere. We may then construct an arbitrary sphere from this circle, and there are many methods for doing so (eg picking the radius) but we will go with the decidedly simplest method.

Choose \(\theta \in (-\frac{\pi}{2},\frac{\pi}{2})\). Then create a cone by projecting rays from each point on the circle at an angle \(\theta\), so that the rays converge on a point. A sphere is defined by its center and its radius, so the set of all spheres is isomorphic to \(\mathbb{R}^3 \times \mathbb{R}^+\). The tip of the cone is the center, and the length from the center to any point on the circle is the radius.

Hence, we have a function \(f: (-\frac{\pi}{2},\frac{\pi}{2}) \to \mathbb{R}^3 \times \mathbb{R}^+\). Since \(\mathbb{R}^n\) is complete, \(f\) is continuous and thus satisfies the intermediate value property. Eventually, the circle will pass through the fourth point, and since it will only pass through this point once, the sphere defined by our four points is unique.

By induction, this result can be extended to every n, as in the title.

This just goes to show the simultaneous power and rigour analysis can bring to the table.

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