Math 321 Homework
September 7, 2011 (due September 9)
1. (a) Prove that, for 0 < a < b and n ∈ N,
bn − an < (b − a)nbn−1 .
(Hint: use the algebraic identity bn −an = (b−a)(bn−1 +bn−2 a+· · ·+an−1 ).)
(b) Given x > 0 and n ∈ N, prove that there is a unique r > 0 such that
rn = x.
(Hint: follow the proof given in class for the case n = 2, using the first part
rn −x
to help establish the two contradictions. In particular, try using ε = nr
n−1
x−rn
and ε = n(r+1)
n−1 .)
2. A sequence (an ) in R is bounded if the set A = {an | n ∈ N} is bounded. It is
nondecreasing (or weakly increasing) if for every n ∈ N, an+1 ≥ an . Prove
that if (an ) is a bounded nondecreasing sequence in R then (an ) converges to
a = sup A.
Nonincreasing sequences are defined analogously. Show that if (an ) is a
bounded nonincreasing sequence in R then (an ) converges to a = inf A.
3. The lim sup (short for limit superior or limit supremum) of a sequence
(an ) in R is
lim sup an = lim sup ak .
n→∞
n→∞
k≥n
and the lim inf of (an ) is
lim inf an = lim
n→∞
n→∞
inf ak .
k≥n
Suppose that (an ) is bounded.
(a) Prove that the lim sup and lim inf of (an ) both exist. (Hint: see the
previous problem.)
(b) Prove that
lim sup an ≤ b
n→∞
if and only if for every ε > 0, an ≤ b + ε eventually (that is, there is an
N ∈ N such that if n ≥ N then an ≤ b + ε).
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(c) Prove that
lim sup an ≥ b
n→∞
if and only if for every ε > 0, an ≥ b − ε infinitely often (that is, for every
k ∈ N there is an n ≥ k such that an ≥ b − ε).
We’ll encounter lim sup and lim inf again in this course. They are important
and useful quantities, partly because they can sometimes serve as substitutes
for a limit when a limit doesn’t exist. Another important property which we’ll
see later is the following: (an ) converges if and only if
lim inf an = lim sup an .
n→∞
n→∞
(This fact can be proved directly from the definitions above, or from parts (b)
and (c) and the corresponding results about lim inf. We’ll see it instead as an
application of another characterization of lim sup and lim inf in terms of cluster
points.)
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