TITLE: Proofs where higher dimension or cardinality actually enabled much simpler proof?
QUESTION [71 upvotes]: I am very interested in proofs that become shorter and simpler by going to higher dimension in $\mathbb R^n$, or higher cardinality. By "higher" I mean that the proof is using higher dimension or cardinality than the actual theorem.
Specific examples for that:

The proof of the 2-dimensional Brouwer Fixed Point Theorem given by by Aigner and Ziegler in "Proofs from the BOOK" (based on the Lemma of Sperner). The striking feature is that the main proof argument is set up and run in $\mathbb R^3$, and this 3-dimensional set-up turns the proof particularly short and simple.
The proof about natural number Goodstein sequences that uses ordinal numbers to bound from above.
The proof of the Finite Ramsey Theorem using the Infinite Ramsey Theorem.

In fact, I would also be interested in an example where the theorem is e.g. about curves, lattice grids, or planar graphs $-$ and where the proof becomes strikingly simple when the object is embedded e.g. in a torus, sphere, or any other manifold.
Are you aware of proofs that use such techniques?

REPLY [3 votes]: Let $P$ be a convex polytope in $\mathbb{R}^d$ with vertices $v_1,\dots,v_n\in \mathbb{Z}^d$. A nice trick that helps to visualize, understand and prove that the number of lattice points in dilations $tP$ $(t\in\mathbb{N})$ is a polynomial in $t$, called the Ehrhart polynomial of $P$, is to add one dimension a consider the cone over $P$:
$$\mathrm{cone}(P)=\{r_1(v_1,1)+\cdots+r_n(v_n,1)\mid r_1,\cdots,r_n\ge0\}\subset\mathbb{R}^{d+1}.$$
Then the dilated polytope $tP\subset\mathbb{R}^d$ corresponds to the intersection of $\mathrm{cone}(P)$ with the hyperplane $\{(x_1,\dots,x_{d+1})\in\mathbb{R}^{d+1}\mid x_{d+1}=t\}$. This allows to work with some generating functions associated to polyhedra which simplify for cones.