Unformatted text preview: Theorem 13.3. Let ( t n ) be a sequence of real numbers. ( a ) If ( t n ) is increasing and unbounded above, then lim s n = + ∞ . ( b ) If ( t n ) is decreasing and unbounded below, then lim t n =-∞ . Cauchy Sequences Deﬁnition 13.4. A sequence of real numbers ( s n ) is said to be a Cauchy sequence , if for any ± > 0, there exists N , such that m,n > N implies | s n-s m | < ± . Before stating the Cauchy convergence criterion, we state two useful lemmas. Lemma 13.5. Every convergent sequence is Cauchy. Lemma 13.6. Every Cauchy sequence is bounded. Theorem 13.7 (Cauchy convergence criterion) . A sequence of real numbers is convergent iﬀ it is Cauchy....
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- Spring '08
- Real Numbers, Mathematical analysis, Limit of a sequence, Cauchy sequence, Cauchy