Advanced Calculus fi..

Advanced Calculus, Fifth EditionTHEOREM 5Ifthenprovided, in the last case, that there is no division by 0.Proof. The function f (x, y) = x + y is continuous for all x and y. Hence byTheorem 4lim f (x,, y,) = lim (x, + y,) = x + y.n+mn+ccThe other rules follow in the same way, with f (x, y) chosen successively as x - y,x . y, xly.THEOREM 6 (Cauchy Criterion) If the sequence s, has the property that forevery 6 > 0 an N can be found such thatIS, -s,I < E forn > N and m > N, (6.10)then s, converges. Conversely, if s, converges, then for every r > 0 an N can befound such that (6.10) holds.Proof. If (6.10) holds, then the sequence s, is necessarily bounded. Indeed, if wechoose a fixed 6 and N according to (6. lo), then for a fixed m > N,for all n greater than N. Hence all but a finite number of the s, lie in this interval,so that the sequence must be bounded. If the sequence fails to converge, then byTheorem 3 the upper limit S and lower limit E must be unequal. Let S - 5 = a. Thenfor infinitely many m and n. Hence Ism - s, I > a/3 for infinitely many m and n, asFig. 6.2 shows. If 6 is chosen as a/3, condition (6.10) cannot be satisfied for any N.This is a contradiction; hence S = E, and the sequence converges.Conversely, if sn converges to s, then, for a given r > 0, N can be chosen so thatIs, - sl < !p for n > N. Hence form > N and n > N,I s ~ - s ~ I = I s ~ - s + s - s ~ I ~ I S ~ - S I + I S - S , ( < f € + ! j ~ = ~ .Accordingly, condition (6.10) is satisfied.Figure 6.2Cauchy condition.