Download Problem Set 3

MATH 104: INTRODUCTORY ANALYSIS
SPRING 2008/09
PROBLEM SET 3
Unlike the previous problem set, in this one you will need to prove your claims rigorously.
1. (a) Prove Bernoulli’s inequality: (1 + x)n ≥ 1 + nx for every real number x ≥ −1 and every
n ∈ N.
(b) Define the sequence (an )n∈N and (bn )n∈N by
1
2n2 + 1
and
b
=
.
n
2n
n2 + 3n
Prove from the definition of limits that
an = 2 −
lim an = 2 = lim bn .
n→∞
n→∞
In other words, given ε > 0, you should produce a corresponding N ∈ N that satisfies the
definition.
2. Let (an )n∈N and (bn )n∈N be sequences of real numbers.
(a) Suppose limn→∞ an = 0 and (bn )n∈N is bounded (but not necessarily convergent). Prove
that limn→∞ an bn = 0.
(b) Suppose (an )n∈N is convergent and (bn )n∈N is divergent. Prove that (an +bn )n∈N is divergent.
3. Let (an )n∈N and (bn )n∈N be sequences of real numbers. Are the following statements true or
false? You need to prove the statement or give a counterexample.
(a) If (an )n∈N is convergent and (bn )n∈N is divergent, then (an bn )n∈N is divergent.
(b) If (an )n∈N and (bn )n∈N are both divergent, then (an + bn )n∈N is divergent.
(c) If (an )n∈N and (bn )n∈N are both divergent, then (an bn )n∈N is divergent.
4. Let (an )n∈N be a sequence of real numbers. Define the sequence (sn )n∈N by
a1 + a2 + · · · + an
sn :=
n
for every n ∈ N.
(a) Prove that if limn→∞ an = a, then limn→∞ sn = a.
(b) Give an example to show that the converse is not always true.
Date: February 26, 2009 (Version 1.0); due: March 4, 2009.
1