Uniform Convergence

Abstract

Theorem 3. Let I be an interval, and let (fn) be a sequence of differentiable functions from I to R. Suppose that the sequence (f ′ n) of derivatives converges uniformly, and that there exists c ∈ I such that the sequence (fn(c)) of values converges. Then (fn) converges pointwise, lim fn is differentiable, and ( lim n→∞ fn )′ = lim n→∞ f ′ n. Theorem 4. Let A ⊂ R, let ∑∞ n=1 fn be a uniformly convergent series of functions from A to R, and let t ∈ A. If each fn is continuous at t, then so is ∑∞ n=1 fn. Theorem 5. Let ∑∞ n=1 fn be a uniformly convergent series of functions from [a, b] to R. If each fn is integrable, then so is ∑∞ n=1 fn, and ∫ b