gen-state-uniform-convergence-precisely-and-explain-why-it-matte-c1f8

State uniform convergence precisely and explain why it matters in 2-3 sentences.

## Uniform Convergence A sequence of functions {fₙ} **converges uniformly** to f on a set S if: For every ε > 0, there exists N ∈ ℕ such that for all n ≥ N and **for all x ∈ S**, |fₙ(x) - f(x)| < ε. ## Why It Matters Uniform convergence preserves key properties that pointwise convergence does not: if fₙ → f uniformly and each fₙ is continuous, then f is continuous; you can interchange limits and integrals (∫ lim fₙ = lim ∫ fₙ); and you can differentiate/integrate term-by-term in series. These properties are essential for rigorous analysis in calculus, differential equations, and Fourier series, where we need guarantees that limiting operations don't destroy structural properties.