1549901369-Elements_of_Real_Analysis__Denlinger_

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120 Chapter 2 11 Sequences



  1. 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.

  2. 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?

  3. Without using Theorem 2.7.4, prove that if a Cauchy sequence has a
    (finite) cluster point L, then it must converge to L.

  4. Prove (without using Theorem 2.7.4) that a Cauchy sequence of integers
    must be eventually constant.

  5. 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.


  1. 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.

  2. (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.

  3. See Exercise 2.6.23.

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