Proofs that every mathematician should know. [closed]

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 mathematicans should know, and why.

Solution 1:

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 $(\sqrt 2^\sqrt 2)^\sqrt 2=(\sqrt 2)^ 2=2$.

Solution 2:

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.

Solution 3:

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

Solution 4:

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