Advanced Calculus fi..

168 Advanced Calculus, Fifth Edition 1SI S2 $3 s BI I I I II - 1 - - - I I Ic*TL kh10Figure 2.25Convergence of bounded monotone sequence.these are said to be strictly increasing or strictly decreasing. In all cases the sequenceis called monotone.A sequence s, is said to be bounded above if for all n, s, 5 B for some numberB and bounded below if for all n, s, > A for some number A; it is said to be boundedif it is bounded above and bounded below. We observe that a monotone increasing(decreasing) sequence bounded above (below) is necessarily bounded.THEOREM AEve'ry bounded monotone sequence converges.Proof. Let the sequence be s,, n = 1,2, ... and let it be increasing, with s, 5 Bfor all n. We then try to find s by locating it to the nearest integer below s, then tothe nearest tenth and so on, as in Fig. 2.25.In detail we let h ( B c h + 1 , c I sl < c + 1 for integers c, h, so that necessarilyc 5 h. Also c ( s, < h + 1 for all n. We consider the finite set of integers c,c + 1 , . . . , h and let p be the largest of these that is attained or exceeded by some s,.Thus p 5 s, < p + 1 for n sufficiently large. We consider the finite set of numbers34%isi 2and let p + (kl 110) be the largest number in this set attained or exceeded by somes,, so that kl is one of the integers 0, 1. . . . ,9, andfor n sufficiently large. Continuing in this way, we obtain a sequence kl , kz, . . . ,k,, ... of integers, each having one of the values 0, 1, ...,9. Lets be the real numberp + 0.klk2 ... k, . a . Then from our construction we have s, ( s for all n; for ifs, > s, then we would have failed to choose one of the k, as large as possible. Also,for each m,S, 2 p + O.kl k2 . - km > s -for all n sufficiently large. Given r > 0, we choose m so large that lo-" < r (m canbe obtained from the decimal expression for 6). Thus for n sufficiently large, s - r