Surprisingly elementary and direct proofs

What are some examples of theorems, whose first proof was quite hard and sophisticated, perhaps using some other deep theorems of some theory, before years later surprisingly a quite elementary, direct, perhaps even short proof has been found?

A related question is MO/24913, which deals with hard theorems whose proofs were simplified by the development of more sophisticated theories. But I would like to see examples where this wasn’t necessary, but rather the theory turned out to be superfluous as for the proof of the theorem. I expect that this didn’t happen so often. [Ok after reading all the answers, it obviously happened all the time!]

Answer

Goedel’s original completeness/compactness proofs in logic are very hard and technical. Modern versions of the proofs are considerably simpler and do not use any sophisticated new theory.

Most proofs of the fundamental theorem of algebra use some topological/homotopical deep(ish) results or some deep(ish) results of complex analysis. In fact, the theorem can be proved using only the definition of complex numbers and absolute value, using very elementary properties of complex numbers, and the entire proof is about half a page long (such a proof is actually a minimum modulus argument but applied to a polynomial so that the computation can be done directly without appeal to the general minimum modulus principle). The proof is extremely elementary.

The proof that Euclide’s fifth postulate is independent of the other axioms of Euclidean geometry by exhibiting models (i.e., the sphere and the hyperbolic plane) where the postulate fails (in different ways) is completely elementary. Certainly all those who tried to prove/disprove the fifth postulate did so while walking on a counterexample. However, the numerous attacks on the problem prior to its settlement in the 20th century were quite sophisticated and laborious. The barrier was conceptual – the understand of the importance of models – not technical.

Brouwer’s topological proof of his fixed point theorem used (I believe) homology and/or the fundamental group. A perfectly elementary proof using Sperner’s Lemma was later discovered (I’m not entirely sure about the chronological order here, so I may be mistaken).

Initial proofs that the higher homotopy groups are all abelian were greatly simplified by the Eckman-Hilton argument (which is completely elementary).

The Mac Lane coherence theorem in category theory is rather technical and is enormously simplified by considering the Yoneda embedding in the 2-categorical setting. All of the machinery was already present, but it required some re-assembly to figure out a rather elementary proof.

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Ittay Weiss

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