Intro to Analysis in-class_Part_13

# Intro to Analysis in-class_Part_13 - 2.1 Cauchy sequences...

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2.1 Cauchy sequences 31 Idea: as a real number, { x n } = lim x n . Then prove: Theorem 2.1.5. A sequence in R has a limit ⇐⇒ it is Cauchy. Proof. Later: § 2.3. For now, pretend it sounds good. PROBLEM: uniqueness. What if two Cauchy sequences tend to the same L ? SOLUTION: deﬁne them to be the same if they do tend to the same L : Deﬁnition 2.1.6. Two Cauchy sequences { x n } and { y n } are equivalent ( { x n } ’ { y n } ) iﬀ ε > 0 , N ε such that n N = ⇒ | x n - y n | < ε THINK: By Thm just above, this means: two Cauchy sequences are equivalent iﬀ they have the same limit. Theorem 2.1.7. Equivalence of Cauchy sequences really is an equivalence relation. Proof. Must show: reﬂexivity, symmetry, transitivity. 1. reﬂexive: { x n } ’ { x n } . ε > 0 , N ε such that n N = ⇒ | x n - x n | = 0 < ε. 2. symmetric: { x n } ’ { y n } = ⇒ { y n } ’ { x n } . Since

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## This note was uploaded on 11/26/2011 for the course MAT 4944 taught by Professor Wong during the Fall '11 term at FSU.

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Intro to Analysis in-class_Part_13 - 2.1 Cauchy sequences...

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