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A while back I saw posted on someone's office door a statement attributed to some famous person, saying that it is an instance of the callousness of youth to think that a theorem is trivial because its proof is trivial.

I don't remember who said that, and the person whose door it was posted on didn't remember either.

This leads to two questions:

(1) Who was it? And where do I find it in print---something citable? (Let's call that one question.)

(2) What are examples of nontrivial theorems whose proofs are trivial? Here's a wild guess: let's say for example a theorem of Euclidean geometry has a trivial proof but doesn't hold in non-Euclidean spaces and its holding or not in a particular space has far-reaching consequences not all of which will be understood within the next 200 years. Could that be an example of what this was about? Or am I just missing the point?

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This is anti-climactic since you (rather quickly!) chose an answer, but what is your definition of "nontrivial theorem"? For example, would Schur's Lemma or Maschke's theorem have counted? – Boyarsky Jun 20 '10 at 2:10
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I also think this should be community wiki (since there really isn't a right answer). – Akhil Mathew Jun 20 '10 at 2:20
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Very often non-trivial theorems become definitions, or new definitions are specifically chosen so that they become trivial. Thereafter, they have trivial proofs. For example, the fact that homology is invariant under homotopy is (almost) trivial once you know singular homology. Even more often, our whole way of viewing math changes so that we get used to some new amazing discovery (as in Joel's example of existence of uncountable sets below) – Ilya Grigoriev Jun 20 '10 at 3:03
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"callousness" or "callowness"? – Yemon Choi Jun 20 '10 at 3:12
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To Boyarsky: he chose the answer because it gave the source of the quote. – Zsbán Ambrus Jun 20 '10 at 11:43

38 Answers 38

The ultimate example that I know of is the Central Limit Theorem, described by Tijms as ``the unofficial sovereign of probability theory''. Incidentally, its significance took time to sink in- it has been forgotten and reproved repeatedly throughout its history.

Classical CLT: Given iid random variables $X_1,X_2,\ldots$ of mean $0$ and variance $1$, the sequence of random variables $\frac{X_1+X_2+\cdots+ X_n}{\sqrt{n}}$ converges in distribution to a normal random variable with mean $0$ and variance $1$.

The proof just Taylor's theorem and the definition of the exponential function (and Lévy's continuity theorem to confirm that the trivial proof indeed implies the theorem statement):

Proof: The Taylor expansion of the characteristic function $Ee^{itX}$ is: $1-t^2/2+o(t^2)$. Plugging in, the characteristic function of $\frac{X_1+X_2+\cdots+X_n}{\sqrt{n}}$ is $\left(1-t^2/2n+o(t^2/n)\right)^n$ which converges to $e^{-t^2/2}$. By Lévy's continuity theorem, convergence of characteristic functions implies convergence in distribution.QED
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However, Levy's theorem that convergence in distribution is equivalent to convergence of the characteristic functions is less trivial. – Jochen Wengenroth Dec 1 '15 at 9:36
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I agree, but given this result the proof is trivial. Editting to make my answer more honest. – Daniel Moskovich Dec 1 '15 at 11:41

The Nielsen-Schreier Theorem : a subgroup of a free group is a free group.

The algebraic proofs are rather complicated, whereas the topological proof is trivial : a group is free if and only if it acts freely on a simplicial tree.

Of course the theory of covering spaces and fundamental groups is hidden somewhere.

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Many theorems of finite group theory have such nature: they are non-trivial but their proofs are not so hard. But in the frame of infinite groups or finite loops, those are challenging problems. Below are some example:

1- a finite group with just two conjugacy classes is $\mathbb{Z}_2$.

2- a non-trivial finite $p$-group has non-trivial center.

3- finite groups have Lagrange property.

4- a finite group in which its all nontrivial proper subgroup have order a fixed prime $p$ has order $p^2$ and so is abelian.

Many theorems of finite dimensional vector spaces are also non-trivial with trivial proofs: the similar theorems are not true for modules or infinite dimensional cases or have hard proofs.

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I am surprised that no one has mentioned Cantor-Schröder-Bernstein Theorem. It certainly is a non-trivial theorem until you see it for the first time. The proof I linked here, I believe, could be considered a trivial one if you draw "the picture" and observe how the constructed bijection maps the elements.

Another example could be Łoś's theorem. The proof is basically going through definition of ultraproducts and carrying out an induction on formulas. It is tedious to write down but at its core a trivial one. Though, I am reluctant to call the theorem itself trivial!

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I agree about Łoś's theorem, but not the Cantor-Schröder-Bernstein Theorem. The latter seems like the opposite: The theorem is a trivial "squeeze theorem" about cardinal numbers, but it took 3 guys to get a correct proof. The proof is not that long, but it's clever and IMHO not "trivial." – Monroe Eskew May 9 '14 at 1:19

Farkas's Lemma and a variety of other theorems of alternatives are fundamental in the theory of optimization. The proof simply couples the Fundamental Theorem of Linear Algebra with the fact that a positive vector and a (nonzero) nonnegative vector in Euclidean space cannot be orthogonal.

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The theorem that differential generalized cohomology is characterized by a differential cohomolgy exact hexagon -- originally asked/conjectured generally and proven for the ordinary case by (Simons-Sullivan 07) -- turns out to follow formally "by stable cohesion" (Bunke-Nikolaus-Völkl 13). A quick review is here: ncatlab.org/schreiber/show/IHP14.

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$$ \int u\,dv = uv - \int v \, du. $$ The whole theory of generalized functions follows, as do lots of other things.

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I downvoted it because I thought it was a wild exaggeration. – Todd Trimble Dec 2 '15 at 3:49
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The principle of integration by parts occurs frequently and all over the place, no question about it, just as does, for example, completing the square or polarization identities, etc. etc. And it occurs in widely varying contexts. But the formula itself doesn't foreshadow all the objects or all the contexts to which it might apply, which vary in sophistication. So, if you say "the whole theory of generalized functions [distributions] follows", then I say no way, that's a wild exaggeration. Does some manifestation of it play an important role there? Of course! – Todd Trimble Dec 2 '15 at 21:26

The proof that the deRham cohomology is equivalent to singular cohomology on a smooth manifold is in some sense trivial: one shows that the de Rham complex is a soft (hence cohomologically trivial) resolution of the constant sheaf, and it is not too hard to show that the cohomology of the constant sheaf is the same as singular cohomology. In a sense, it just follows from "abstract nonsense" about derived functors being computable from acyclic resolutions and the fact that soft resolutions are acyclic (a partition of unity argument). But it is certainly a nontrivial theorem.

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So the proof of Serre's GAGA theorems is trivial, granting analytic and algebraic finiteness and ampleness theorems. The proof of Mordell-Weil is trivial, granting the theory of abelian varieties, height functions, and general finiteness theorems of Galois cohomology. If one does the hard foundational work beforehand, then yes, what remains is trivial. To make the deRham theorem useful, we need properties of the isomorphism. So a test: can you prove the integration map trivialization of top-degree cohomology on a compact oriented manifold matches the one defined in singular cohomology? – Boyarsky Jun 20 '10 at 2:51
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Unfortunately not, because I know nothing about singular cohomology. – Akhil Mathew Jun 20 '10 at 3:26
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Akhil, no worries: now you have a good exercise to keep in mind as you learn more about singular cohomology. You probably won't find it proved in any book (I never did), but eventually you'll figure it out for yourself. At least it gives you more appreciation for the subtlety of the deRham isomorphism. (By the way, the compatibility with cup products is another good one, but that is elegantly handled in Godement's sheaf theory book via "general nonsense" with pairings of resolutions.) There's also the matter of cohomology with compact supports... – Boyarsky Jun 20 '10 at 3:44
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Wow, I thought I was the only person in the world who learned sheaf cohomology before learning any other type of cohomology theory! I remember that the first diagram I ever chased was the diagram giving the long exact sequence in cohomology associated to a short exact sequence of sheaves (my teacher told me that it would be more instructive to prove it by myself <grin>). – Andy Putman Jun 20 '10 at 19:22
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Akhil, the book of Greenberg & Harper is short & sweet (with nice exercises). Their discussion of orientation is curious since use an "orientation sheaf" but don't have the general notion of sheaf and so get stuck in some contortions. Anyway, you can get a .djvu file of Munkres' book from our Russian friends at extracoder.com/genesis/0072.html (look for item 72583 in the numbering of the left column). And .djvu of Greenberg & Harper is at rapiddigger.com/download/… – Boyarsky Jun 21 '10 at 3:27

protected by François G. Dorais May 8 '14 at 21:53

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