By Hugo D. Junghenn

ISBN-10: 148221928X

ISBN-13: 9781482219289

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1 Limits of Sequences Simply stated, a sequence in a set E is a function from N to E. It is more instructive, however, to think of a sequence as an infinite ordered list of members of E. The list may be written out, for example, as a1 , a2 , . . , an , . . or abbreviated by {an }∞ n=1 or simply by {an }. A sequence usually starts with the index 1, although this is not necessary, 0 being a common alternative. The set E in the definition of sequence is arbitrary. However, for Part I of the book, we consider only numerical sequences, that is, sequences contained in R.

A sequence {an } in R is said to converge to a real number a, written an → a or lim an = lim an = a, n n→+∞ if for each ε > 0 there exists N ∈ N such that |an − a| < ε, (a − ε < an < a + ε), for all n ≥ N. If no such real number a exists, then the sequence is said to diverge. 1: Convergence of a sequence to a It follows immediately from the definition that an → a iff the terms of sequence eventually lie in any open interval containing a. The definition also implies that an → a iff |an − a| → 0.

To find x, let n → ∞ in the equation xn+1 = a+ xn − a to obtain x = a+ x − a. This has solutions x = a and x = b. Since {xn } is increasing, x = b. Similarly, y = a. 4 Example. We use the monotone sequence theorem to show that the sequence {(1 + 1/n)n } converges. 5) and the inequality k! ≥ 2k−1 (easily established by induction), n n (1 + 1/n) = k=0 n 1/nk k n =2+ (1 − 1/n)(1 − 2/n) · · · (1 − (k − 1)/n)/k! k=2 n ≤2+ 1/2k−1 . k=2 n Since the sum in the last inequality is ≤ 1, {(1 + 1/n) } is bounded above by 3.

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A course in real analysis by Hugo D. Junghenn

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