Proofs that every mathematician should know.

Question

There are mathematical proofs that have that "wow" factor in being elegant, simplifying one's view of mathematics, lifting one's perception into the light of knowledge, etc.

So I'd like to know what mathematical proofs you've come across that you think other mathematicians should know, and why.

Answer 1

I think every mathematician should know the following (in no particular order):

1. Pythagorean Theorem.
2. Summing $\sum_{k = 1}^{n} k$ using Gauss' triangle trick.
3. Irrationality of $\sqrt{2}$ by proof without words.
4. Niven's proof of the irrationality of $\pi$.
5. Uncountability of the Reals by Cantor's Diagonal Argument.
6. Denumerability of the Algebraics by Heights and Counting Roots.
7. Infinitude of primes by both Euclid's proof and Euler's proof.
8. Constructibility of the Regular 17-gon by Gauss' explicit construction.
9. Binomial Theorem by Induction.
10. FLT $n = 4$ by Fermat's Infinite Descent.
11. Every ED is a PID is a UFD.
12. The $\lim_{n \to \infty} (1 + \frac{1}{n})^{n} = e$ by L'Hôpital's Rule.
13. Pick's Theorem by reduction to triangles and squares.
14. Fibonacci numbers in terms of the Golden Ratio by recurrence relations.
15. $\mathbb{R}^{n}$ is a metric space in more than one way.
16. Euler's Formula $e^{i \theta} = \cos \theta + i \sin \theta$ by differentiation.
17. Summing $\sum_{k \geq 1} \frac{1}{k^{2}}$ by Fourier series.
18. Quadratic reciprocity by Eisenstein's proof (counting lattice points).
19. $(\mathbb{Z}/n \mathbb{Z})^{\times}$ is a group (of units) for $n \in \mathbb{N}$, and $\mathbb{Z} / p \mathbb{Z}$ is a field for prime $p$.
20. Euler's formula $v - e + f = 2$ for planar graphs.
21. Fundamental Theorem of Algebra by Liouville's Theorem.

This is, of course, my opinion....

NB: When I write "by X" above (where X is a specific methodology or theorem), I suggest that one learn by that route (as opposed to another perhaps easier route), because of the specific pedagogical benefit.

Answer 2

I'm particularly fond of Ramsey's Theorem.

Answer 3

Proofs from THE BOOK is a brilliant compilation of such beautiful succinct proofs.

Answer 4

I would say the proof of the Brouwer Fixed Point Theorem for $D^n$ using the fact that $H_n(S^n) \cong \Bbb{Z}$ and $H_n(D^n) = 0$ is nice. The idea of the proof by contradiction that if for no $x$ is $f(x) = x$, we can draw a straight line through these points, that for me was very elegant.

Answer 5

Gödel's theorem was definitely a "wow" for me.

Also interesting, the proof around the non-Enumerability of $\mathbb{R}$.

In the same area, the Fact that $\mathbb{Q}$ is a dense subset of $\mathbb{R}$, despite the fact that $\mathbb{Q}$ is numerable while $\mathbb{R}$ is not. It kind of suggests that $n(\mathbb{Q}) = n(\mathbb{R}-\mathbb{Q})$ while at the same time $\mathbb{R}-\mathbb{Q}$ is infinitely bigger than $\mathbb{Q}$...

Church numerals if you're into computer science.

Answer 6

Here is one strategy for proving Fermat's Last Theorem: suppose you could show that $x^n + y^n = z^n$ has no nontrivial solutions ${}\bmod p$ for infinitely many primes $p$. Then any nontrivial solution over $\mathbb{Z}$ necessarily reduces to a nontrivial solution mod a sufficiently large prime, so you've proven FLT.

Unfortunately, this is false: for fixed $n$, the Fermat equation has nontrivial solutions ${}\bmod p$ for all sufficiently large primes $p$! This was first proven by Schur (I am told), and the proof uses Ramsey theory and very little actual number theory. I think this proof teaches the following valuable lessons:

• What seems like a problem in one field might be best thought of as a problem in another.
• Sometimes the way to solve a problem is to ignore a lot of its structure.

Answer 7

The Weyl Character formula is an excellent example of a deep result with a clever proof. The result states (in one form) that the irreducible characters of a compact, connected Lie group are parametrized uniquely by their heighest weight vectors! The intuition is that characters on a compact, connected Lie group $G$ are class functions on $G$ and their restrictions to a maximal torus of $G$ are $W$-invariant functions on $T$ where $W$ is the Weyl group of $T$. The $W$-invariance of a character on $T$ allows you to parametrize it by a heighest weight vector using the theory of roots and weights. However, the clever point of the proof is that one studies $W$-anti-invariant functions on $T$ rather than $W$-invariant functions on $T$! The quotient of two $W$-anti-invariant functions on $T$ is a $W$-invariant function on $T$. I think that this is a deep and extremely important result in mathematics with a clever proof.

Answer 8

The full classical proof of the classification theorem of compact surfaces has always been-and remains-one of my favorite proofs. Despite it's tediousness, it demonstrates to the beginner how important it is to be able to prove results constructively,using very little beyond the definitions of a surface and the fundamental group.

Answer 9

I would have to include (at least) one of the proofs available for quadratic reciprocity. My personal preference would be for the proof due to Eisenstein presented in Ireland and Rosen, but there are so many others to choose from.

A second one I would include would be Minkowski's lattice point theorem, as proved in Hasse's "Number Theory".

Answer 10

Here is my favourite "wow" proof .

Theorem
There exist two positive irrational numbers $s,t$ such that $s^t$ is rational.
Proof
If $\sqrt2^\sqrt 2$ is rational, we may take $s=t=\sqrt 2$ .
If $\sqrt 2^\sqrt 2$ is irrational , we may take $s=\sqrt 2^\sqrt 2$ and $t=\sqrt 2$ since $(\sqrt 2^\sqrt 2)^\sqrt 2=(\sqrt 2)^ 2=2$.

Answer 11

.... and of course the neat proof that $$ \int_{-\infty}^\infty e^{-x^2}\,dx=\sqrt{\pi}. $$

Answer 12

Cantor's Theorem: There is no surjection from $A$ onto $\mathcal P(A)$.

Answer 13

Perhaps geometric and algebraic proofs of the fundamental theorem of calculus.

Answer 14

The proof of the Fundamental Theorem of Algebra via Liouville's theorem is short and sweet.

Answer 15

As a beginner (and far from being a mathematician) there are two proofs that I have come across that I would say, for me, were "symphonic" capers of some areas of study - showing what can be done with material you've studied.

An additional point is that the actual presentation of the proofs themselves were instructive in their elegance. Sort of like a virtuoso performer.

The Stone-Weierstrass Theorem - Vaughan Jones https://sites.google.com/site/math104sp2011/lecture-notes ({Wayback Machine](https://web.archive.org/web/20171104025413/https://sites.google.com/site/math104sp2011/lecture-notes))

The Sylow Theorems - Benedict Gross http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra (Wayback Machine)

Answer 16

The ultrafilter proof of Tychonoff's theorem.

The proof is simple, show the power of working with filters and incorporats a good deal of what "everyone should know about compactness".

The strategy-stealing argument for why the first player can force a win in hex.

The argument is simple, elegant, clever and there is essentially no effort in learning it.

The proof of Zorn's lemma by way of ordinals.

Too many people believe that Zorns lemma is an inherently incomprehensible black box. It is not.

Heine-Borel by "induction."

The argument is very neat and shows exactly where the completeness of $\mathbb{R}$ matters.

The visual argument for finding the area of a circle, given radius and circumference.

It's simply beautiful.

Answer 17

I personally believe some of the proofs of Pythagoras' theorem can be both beautiful and elegant, though it is unfortunate that it is not taught in school (at least as far as I am aware).

Take any square with sides of length $x+y$. Then $x$ units from each corner, connect to the next corner, again $x$ units away. Call this distance $z$. Therefore you have a square with side length $x+y$ with four triangles with base and height $x$ and $y$ and a smaller square in the middle with a side length of $z$.

$$(x+y)^2=4\frac{1}{2}xy+z^2$$ $$x^2+y^2+2xy-2xy=z^2$$ $$x^2+y^2=z^2$$

Answer 18

It is probably not something which everybody should know, nevertheless it is simply beautiful!

The stable Hurewicz theorem using that the sphere spectrum is a compact generator of the stable homotopy category. In particular Serre's theorem that the rational stable homotopy groups of spheres are trivial for degrees bigger than 1.

Answer 19

$$2+2=4$$

Of course I am talking in the context of Peano Axioms (or some other reasonable theory of arithmetics).

Indeed most mathematicians could come up with the proof in a matter of minutes, after seeing the axioms, the trick of course is to understand what is there to prove here anyway?

We defined $+$ by induction. We denote by $2=S(1)$ and $4=S(S(S(1)))$. Now we need to prove that the terms $S(1)+S(1)$ and $S(S(S(1)))$ are equal, because there is no axiom tell us that directly.

Answer 20

When I did my first analysis course I found the proof the Lebesgue differentiation theorem using maximal functions and covering lemma arguments to be very beautiful.

Answer 21

I really like the simple and nice proof of the 5-color theorem (i.e. that for every planar graph there exists a vertex coloring with not more than 5 colors) and how surprisingly difficult it is to proof the sharper 4-color theorem.

Answer 22

Euclid's proof. Very simple, very elegant

Theorem

There are more primes than found in any finite list of primes.

Proof

Call the primes in our finite list $p_1, p_2, ..., p_r$. Let P be any common multiple of these primes plus one (for example, $P = p_1p_2...p_r+1$). Now P is either prime or it is not. If it is prime, then P is a prime that was not in our list. If P is not prime, then it is divisible by some prime, call it p. Notice p can not be any of $p_1, p_2, ..., p_r$, otherwise p would divide 1, which is impossible. So this prime p is some prime that was not in our original list. Either way, the original list was incomplete.

Answer 23

My all time favorite proof?

Furstenberg's proof of the infinity of primes by point-set topology. I know, a lot people think it's not a big deal. I think:

a) It's an immensely clever way to use point set topology to prove a result in a seemingly unrelated field, namely number theory.

b) I used it as the beginnings of my first research in additive number theory; looking to generalize this result to create similar proofs of results for sumsets and arithmetic progressions. Sadly, my health failed again, but I hope to return to this research soon.