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Additional info for Schaums Outline of Advanced Calculus (2nd Edition) (Schaum's Outlines Series)
Example text
Hence, by the Bolzano–Weierstrass theorem there is at least one limit point, say a. If a is the only limit point, we have the desired proof and lim un ¼ a. 1 Suppose there are two distinct limit points, say a and b, and suppose b > a (see Fig. 2-1). By definition of limit points, we have jup À aj < ðb À aÞ=3 for infinnitely many values of p juq À bj < ðb À aÞ=3 for infinitely many values of q ð1Þ ð2Þ b_a 3 b_a 3 a b Then since b À a ¼ ðb À uq Þ þ ðuq À up Þ þ ðup À aÞ, we have jb À aj ¼ b À a @ jb À uq j þ jup À uq j þ jup À aj Fig.
N v1 þ v2 þ Á Á Á þ vn ¼ 0 if lim vn ¼ 0. 1 v 1 þ v 2 þ Á Á Á þ v n @ jv1 þ v2 þ Á Á Á þ vP j þ jvPþ1 j þ jvPþ2 j þ Á Á Á þ jvn j n n n Since lim vn ¼ 0, we can choose P so that jvn j < =2 for n > P. 1 ð1Þ Then jvPþ1 j þ jvPþ2 j þ Á Á Á þ jvn j =2 þ =2 þ Á Á Á þ =2 ðn À PÞ=2 < ¼ < n n n 2 ð2Þ After choosing P we can choose N so that for n > N > P, jv1 þ v2 þ Á Á Á þ vP j < n 2 Then using (2) and (3), (1) becomes v 1 þ v 2 þ Á Á Á þ v n < þ ¼ 2 2 n ð3Þ for n > N thus proving the required result.
Where each interval is contained in the preceding one and lim ðan À bn Þ ¼ 0. Such intervals are called nested intervals. 1 We can prove that to every set of nested intervals there corresponds one and only one real number. This can be used to establish the Bolzano–Weierstrass theorem of Chapter 1. ) CAUCHY’S CONVERGENCE CRITERION Cauchy’s convergence criterion states that a sequence fun g converges if and only if for each > 0 we can find a number N such that jup À uq j < for all p; q > N. This criterion has the advantage that one need not know the limit l in order to demonstrate convergence.