Page 42 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 2] SEQUENCES 33

2.23. Prove the Bolzano–Weierstrass theorem (see Page 6).
Suppose the given bounded infinite set is contained in the finite interval ½a; bŠ.Divide this interval into
two equal intervals. Then at least one of these, denoted by ½a 1 ; b 1 Š,contains infinitely many points.
Dividing ½a 1 ; b 1 Š into two equal intervals, we obtain another interval, say, ½a 2 ; b 2 Š,containing infinitely
many points. Continuing this process, we obtain a set of intervals ½a n ; b n Š, n ¼ 1; 2; 3; ... ; each interval
contained in the preceding one and such that
2
b 1 a 1 ¼ðb aÞ=2; b 2 a 2 ¼ðb 1 a 1 Þ=2 ¼ðb aÞ=2 ; ... ; b n a n ¼ðb aÞ=2 n
from which we see that lim ðb n a n Þ¼ 0.
n!1
This set of nested intervals, by Problem 2.22, corresponds to a real number which represents a limit
point and so proves the theorem.

CAUCHY’S CONVERGENCE CRITERION

2.24. Prove Cauchy’s convergence criterion as stated on Page 25.
Necessity. Suppose the sequence fu n g converges to l. Then given any > 0, we can find N such that
ju p lj < =2for all p > N and ju q lj < =2for all q > N
Then for both p > N and q > N,we have
ju p u q j¼jðu p lÞþðl u q Þj @ ju p ljþjl u q j < =2 þ =2 ¼
Sufficiency. Suppose ju p u q j < for all p; q > N and any > 0. Then all the numbers u N ; u Nþ1 ; .. .
lie in a finite interval, i.e., the set is bounded and infinite. 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 u n ¼ a.
n!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
ju p aj < ðb aÞ=3for infinnitely many values of p _ _
b a b a
ð1Þ
ju q bj < ðb aÞ=3for infinitely many values of q ð2Þ 3 3
a b
Then since b a ¼ðb u q Þþðu q u p Þþðu p aÞ,we have
Fig. 2-1
jb aj¼ b a @ jb u q jþju p u q jþju p aj ð3Þ
Using (1) and (2)in(3), we see that ju p u q j > ðb aÞ=3for infinitely many values of p and q,thus
contradicting the hypothesis that ju p u q j < for p; q > N and any > 0. Hence, there is only one limit
point and the theorem is proved.

INFINITE SERIES

2.25. Prove that the infinite series (sometimes called the geometric series)

1
X
2 n 1
ar
a þ ar þ ar þ ¼
n¼1
(a) converges to a=ð1 rÞ if jrj < 1, (b) diverges if jrj A 1.
2
Let S n ¼ a þ ar þ ar þ þ ar n 1
2
Then rS n ¼ ar þ ar þ þ ar n 1 þ ar n
Subtract, ð1 rÞS n ¼ a ar n