121616949-math.253 - 10.1 is increasing and n 1 n = i=1...

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10.1 Sequences 239 is increasing, and n + 1 n i =1 = 2 1 , 3 2 , 4 3 , 5 4 , . . . is decreasing. A sequence is bounded above if there is some number N such that a n N for every n , and bounded below if there is some number N such that a n N for every n . If a sequence is bounded above and bounded below it is bounded . If a sequence { a n } n =0 is increasing or non-decreasing it is bounded below (by a 0 ), and if it is decreasing or non- increasing it is bounded above (by a 0 ). Finally, with all this new terminology we can state an important theorem. THEOREM 10.12 If a sequence is bounded and monotonic then it converges. We will not prove this; the proof appears in many calculus books. It is not hard to believe: suppose that a sequence is increasing and bounded, so each term is larger than the one before, yet never larger than some fixed value N . The terms must then get closer and closer to some value between a 0 and N . It need not be N , since N may be a “too-
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