TITLE: Theorems with many distinct proofs
QUESTION [36 upvotes]: I was told that whenever one learns a new technique, it is a good idea to see if one can prove a well-known theorem using the new technique as an exercise. I am hoping to build a list of such theorems to test my technique.
My Question. What are some theorems with a.) many different proofs, or b.) proofs using strikingly different techniques from mathematics?
Theorems with multiple proofs:

Fundamental theorem of algebra
Fundamental theorem of arithmetic
Existence of Jordan normal form. (Standard minimal polynomial proof, PID proof, Terry Tao's proof, proof using complex analysis)
(Related to prev.) Cayley-Hamilton theorem
Spectral theorem
Quadratic reciprocity law
Pólya's recurrence theorem
Basel problem
Stirling's formula

(Arguably:)

Uniform boundedness principle. (Pf 1. Baire category, Pf 2. Gliding hump)
Brouwer fixed point theorem (although all the proofs I know of boil down to showing that the ball is not homeomorphic to the sphere)

I think the Pythagorean theorem also satisfies my description, but it is a bit too elementary.
Clarification 1: In order to prevent this question from being "what are some theorems that have higher level generalizations," a trivial specialization of a harder theorem (e.g. Hilbert space Pythagorean theorem implying Euclidean Pythagorean theorem) will not be considered a new proof, for the purposes of this question.
Clarification 2: This question differs from this similar stackexchange question in the following sense. The stackexchange post asked for a.) very elementary theorems (lower level undergrad), and b.) short proofs. My question asks for theorems from all levels of mathematics (up to say third year graduate level), which are (preferably) central to the theory.

REPLY [2 votes]: The local monodromy theorem is a fundamental result on the topology of families of complex algebraic varieties. Let $X \rightarrow B$ be a smooth projective family of complex algebraic varieties over a smooth complex algebraic curve $B$, complement of a finite set $S$ of points in a compact curve $\overline{B}$. Then the local monodromy theorem states that for every point $s \in S$, the monodromy action around $s$ on the cohomology (with rational coefficients) of the fibers of $X \rightarrow B$ is quasi-unipotent (i.e. all its eigenvalues are roots of unity).
There are many proofs of this theorem, using very different ideas (a summary is given in section 3 of https://www.ams.org/journals/bull/1970-76-02/S0002-9904-1970-12444-2/S0002-9904-1970-12444-2.pdf ):

there is a "global geometric" proof due to Landman, using Lefschetz pencils and induction on the dimension.

there is a "local geometric" proof due to Grothendieck in the algebraic setting and Clemens in the Kähler setting, based on a local study of vanishing cycles.

there is a "complex hyperbolic geometric" proof due to Borel using the differential geometric fact that moduli spaces of polarized Hodge structures are negatively curved.

there is a proof by Brieskorn using the regularity of the Gauss-Manin connection and the Gelfond-Schneider theorem on transcendental numbers.

there is an "arithmetic" proof due to Grothendieck based on étale cohomology, reduction to positive characteristic, and general properties of l-adic Galois representations.

there is another "arithmetic" proof due to Katz, using the study of the Gauss-Manin connection in positive characteristic.