Prove. Compile. Conquer.

Provide a formal LaTeX proof that the sum of the first n positive integers equals n(n+1)/2. Use induction or a combinatorial argument. Full amsmath/amsthm environments expected.

Reproduce Euclid's proof that there are infinitely many prime numbers. Your proof must be typeset with theorem and proof environments, include a clear contradiction argument, and compile without errors.

Formalize Cantor's diagonal argument proving the uncountability of the real numbers. Include a constructed diagonal number and the contradiction with assumed enumeration. Use array or tabular environments for the diagonal.

Prove both parts of the Fundamental Theorem of Calculus. Part I: if F is an antiderivative of f, then ∫ₐᵇ f(x)dx = F(b)−F(a). Part II: d/dx∫ₐˣ f(t)dt = f(x). Use proper ε-δ formalism.

Prove by contradiction that √2 is irrational. Formally define rational numbers, assume √2 = p/q in lowest terms, and derive the contradiction. Use amsmath for algebraic steps.

Derive e^{iπ} + 1 = 0 from the Taylor series of e^x, sin(x), and cos(x). Show each series expansion explicitly and combine them. Advanced LaTeX formatting required.