By Joseph R. Lee
ISBN-10: 0124407501
ISBN-13: 9780124407503
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717Γ an = - + sin — n 2 suggests still another kind of limit for the sequence {an}. Obviously an infinite number of points in the sequence cluster about 0, but 18 II. Sequence Spaces and Infinite Series that is neither the limit, the limit superior, nor the limit inferior. 6 DEFINITION The point a is a limit point, or cluster point, of the sequence {an\ if every interval (a — e, a + e) contains points of {an} other than a itself. (We leave for an open argument the question of whether the sequence 0, 1, 0, 1, 0, 1, .
8 II. Sequence Spaces and Infinite Series EXAMPLE The harmonic series ΣΓ=ι Iß diverges. For, writing out the first few terms of the series shows that Ϊ 2 iii = 1 = -, A 3 A2 = -, „ 4 A, > -, , 5 As> 6 „ Au > - , . . -, Then, limn^« A n = oo . =ι α* and J X i ¿>fc be series of nonnegative terms with ak 2 COMPLETENESS AXIOM In the real number system every nonempty set of numbers bounded above has a least upper bound, or supremum. It can then be seen that every nonempty set bounded below has a greatest lower bound, or infimum. 414, . . < a; and, for every e > 0, there are numbers in the sequence greater than a — e. Then a is written Λ/2. 3 THEOREM Every bounded monotone sequence of numbers converges. real Proof We prove this for the case that {an} is nondecreasing and leave the others as problems.