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:
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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.
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The proof about natural number Goodstein sequences that uses ordinal numbers to bound from above.
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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?
Best Answer
Whitney's theorem is an example of this. To prove the weak version (i.e. embedding a manifold $M^n$ in $\mathbb{R}^{2n +1}$), you start by using a partition of unity to embed $M^n$ into $\mathbb{R}^{N}$ where $N$ is very large. This is relatively easy to do when $M^n$ is compact and takes a little bit of thought otherwise, but is significantly easier than trying to get an embedding in a lower dimension from scratch. You can then use transversality arguments to show that a generic projection map preserves the embedding of $M^n$ to cut down $N$ until you get to $\mathbb{R}^{2n +1}$.
To get the strong version of the theorem (embedding $M^n$ in $\mathbb{R}^{2n}$), there is another insight needed, which is using Whitney's trick to get rid of double points. As such, it's really the weak version where the high-dimensionality approach is used.