120 Chapter 2 11 Sequences
- Suppose that {xn} is a sequence such that Vn EN, [xn+1 -xn[ <en, for
some constant 0 < C < 1. Prove that { Xn} is a Cauchy sequence, hence
converges. - Find an example of a sequence {xn} such that lim [xn+l - Xn[ = 0 and
n-+oo
lim Xn = +oo. In what sense is this a caution to those who would use a
n-+oo
calculator or computer to conclude whether a sequence converges? - Without using Theorem 2.7.4, prove that if a Cauchy sequence has a
(finite) cluster point L, then it must converge to L. - Prove (without using Theorem 2.7.4) that a Cauchy sequence of integers
must be eventually constant. - Suppose {xn} and {yn} are Cauchy sequences, and r ER Without using
Theorem 2.7.4,
(a) prove that {xn + Yn}, {xn - Yn}, {rxn}, and {XnYn} are Cauchy
sequences, but { ~= } is not necessarily a Cauchy sequence, even if
Vn E N, Yn =I-0.(b) find a condition on {Yn} that will guarantee that {~=}is a Cauchy
sequence, and prove your claim.- A sequence { Xn} is said to be a contractive sequence if ::J some constant
c, 0 < c < 1 3 Vn E N, [xn+2 - Xn+i I :::; c [xn+l - Xnl· Prove that a
contractive sequence must be a Cauchy sequence, and hence converges. - (Project) Recursive Arithmetic Means:^15 Let a =I-b be arbitrary
real numbers, and define the sequence { Xn} by
Xn+l + Xn
x 1 =a, X2 = b, and Vn EN, Xn+2 =
2
.
That is, each new term beginning with the third is the average of the two
previous terms.
(a) Prove that {xn} converges by proving that it is a contractive se-
quence.
(b) Prove that Vn EN, Xn+i + ~Xn = b +~a.
( c) Use (b) and the algebra of limits to find lim Xn. Are you surprised
n-+oo
by this answer? Notice that if you interchanged a and b the answer
would be different. - See Exercise 2.6.23.