1549901369-Elements_of_Real_Analysis__Denlinger_

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72 Chapter 2 • Sequences

'17. Prove that each of the following sequences diverges:
(a) {4n-5} (b) { J3n + 1}

( c) { 2~ 3 } ( d) { ~: : ~ }

(Hint: in each case, show that the sequence is unbounded.)


  1. Prove Theorem 2.2.16.





    1. Justify the following assertion: Vn 0 E N, deleting or altering any or all
      terms of a sequence {xn} before the n 0 th one will affect neither its con-
      vergence/divergence, nor its limit in case of convergence.





  1. Prove or disprove (if false, give a counterexample):


(a) If {xn} and {yn} are both bounded above, then so is their sum
{xn + Yn}·
(b) If { Xn} and {Yn} are both bounded above, then so is their difference
{xn - Yn}·
( c) If { Xn} and { Yn} are sequences of nonnegative real numbers that are
bounded above, then so is their product {XnYn}·
(d) If {xn} and {Yn} are sequences of positive real numbers that are
bounded above, then so is their quotient { ~= }.

/ 21. Prove that if an ---. u and bn ---. v, then max{ an, bn} ---. max{ u, v} and
min{ an, bn} ---. min{ u, v }. [See Exercise 1.2-B.6.]



  1. Suppose an+ bn---. u E IR and an -bn---. v E IR. Prove that {an}, {bn},
    and { anbn} all converge, and find their limits.

  2. Rearrangements: We say that a sequence {Yn} is a rearrangement
    of a sequence {xn} if there is a 1-1 correspondence f : N---. N such that
    \:In EN, Yn = Xf(n)· Suppose {yn} is a rearrangement of {xn}· Prove that
    {yn} converges iff {xn} converges, and when they converge they have the
    same limit.

  3. (Project) Arithmetic Means: For a given {xn} we define its sequence
    X1 +x2 + · · · +x
    of arithmetic means to be {o-n}, where O"n = n.
    n
    (a) Prove that if Xn---. L, then O"n---. L.
    (b) Prove that the converse of (a) is false, by finding a sequence { Xn}
    such that {un} converges but {xn} does not.

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