Intro to Analysis in-class_Part_53

# Intro to Analysis in-class_Part_53 - 7.3 Uniform...

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Unformatted text preview: 7.3 Uniform convergence 111 7.3.2 Criteria for uniform convergence Theorem 7.3.5 (Cauchy Criterion) . { f n } converges uniformly on I iff ∀ ε > , ∃ N m,n ≥ N = ⇒ | f n ( x )- f m ( x ) | < ε, ∀ x. Proof. HW. ( ⇒ ), use Δ ineq. ( ⇐ ), use pointwise Cauchy Crit to obtain the limit f . Theorem 7.3.6 (Weierstrass M-test) . Let { f n } be defined on I and satisfy | f n ( x ) | ≤ M n . If ∑ M n converges, then ∑ f n ( x ) converges uniformly on I . Proof. Fix ε > 0. Then fl fl fl fl fl m X i = n f i ( x ) fl fl fl fl fl ≤ m X i = n | f i ( x ) | ≤ m X i = n M n < ε, for n,m >> 1, because ∑ M n converges. The result follows from the previous thm. Example 7.3.9. ∑ cos nx n 2 converges uniformly on R . Note that fl fl cos nx n 2 fl fl ≤ 1 n 2 and ∑ 1 n 2 converges. In fact, ∑ cos f n ( x ) n 2 converges uniformly on R for any arbitrary f n ( x ). 7.3.3 Continuity and uniform convergence Theorem 7.3.7. A uniform limit of continuous functions is continuous.A uniform limit of continuous functions is continuous....
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Intro to Analysis in-class_Part_53 - 7.3 Uniform...

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