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.)- Prove Theorem 2.2.16.
- 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.
- Justify the following assertion: Vn 0 E N, deleting or altering any or all
- 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.]
- Suppose an+ bn---. u E IR and an -bn---. v E IR. Prove that {an}, {bn},
and { anbn} all converge, and find their limits. - 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. - (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.