Sterling K. Berberian's A First Course in Real Analysis (Undergraduate Texts in PDF

Xn (n E lP') . In the notation (Xn) , the integers n are called the indices. 2. Definition. A sequence (an) in IR is said to be increasing if al ::; a2 ::; a3 ::; . . , that is, if an ::; an+! for all n E lP' ; strictly increasing if an < an+! for all n; decreasing if al ~ a2 ~ a3 ~ ... ; and strictly decreasing if an > an+l for all n. A sequence that is either increasing or decreasing is said to be monotone; more precisely, one speaks of sequences that are 'monotone increasing' or 'monotone decreasing'.

Cauchy's Criterion 49 (b) ~ (a): Assuming (b), let 's show first that the sequence (an) is bounded. Choose an index M such that lam -ani < 1 for all m, n 2: M. Then, for all n 2: M , therefore the sequence (an) is bounded; explicitly, if then Ian I :s; r for all n. 9); we will show that an --+ a. Let E > O. 2); while we're at it, we can require that k 2: K. Then, for all n 2: N , by (*) and (**). 1). (> --+ a The interest of Cauchy's criterion is that the condition in (b) can often be verified without any knowledge as to the value of the limit of the sequence (a virtue shared by the convergence criterion for monotone sequences mentioned at the beginning of the section).

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