gen-in-3-4-sentences-explain-uniform-convergence-state-any-key-f-1c49

In 3-4 sentences, explain uniform convergence. State any key formula.

# Uniform Convergence A sequence of functions {fₙ} converges **uniformly** to a function f on a set S if the convergence rate is the same across all points in S. Unlike pointwise convergence (where the rate can vary by location), uniform convergence requires that for any ε > 0, we can find a single N (independent of x) such that |fₙ(x) - f(x)| < ε for all n > N and all x ∈ S. **Key formula:** $$\lim_{n \to \infty} \sup_{x \in S} |f_n(x) - f(x)| = 0$$ This is equivalent to saying the maximum deviation between fₙ and f vanishes as n → ∞.