Proofs

Ὅσα τὰ αὐτὰ τῷ αὐτῷ ἴσα, καὶ ἀλλήλοις ἐστὶν ἴσα.

“Things equal to the same thing are equal to one another.”

Let 𝑀 be the set of mathematical ideas (proofs, lemmas, theorems). \forallt \geq 0: I(t) \in Learners \land Love(ℳath) = ⊤ \land Ambition(I) \neq “claim-to-mastery”. Purpose: maximize Growth(I,t) subject to Ethics \land Clarity \land Rigor. Method: Write(I, proofs) \Rightarrow (Practice ↑) \land (Insight ↑) \land (Error \to Lesson). Therefore: lim t\to\infty Skill(I,t) ↑, while Humility(I,t) \geq constant ＞ 0.

Irrationality of √2

Four distinct proofs

Proof of Cauchy–Schwarz Inequality

A classical algebraic argument

Bézout’s Identity

Euclid algorithm ⇒ ax + by = gcd(a,b)

Infinitude of Primes

Euclid’s classic contradiction proof

AM–GM Inequality (Two Variables)

(a+b)/2 ≥ √(ab)

Sum of First n Odd Numbers

1+3+…+(2n−1) = n²

Fermat’s Little Theorem

a^(p−1) ≡ 1 (mod p)

Handshaking Lemma

∑ deg(v) = 2E