Due: Friday, Feb 22, 2008
Math 360 - Advanced Calculus / Problem Set 5
Sequences
1) Finish the proof of the assertion from the class: Every monotone bounded sequence of real numbers (xn)n
is convergent. More precise:
a) If (xn)nis increasing, then lim
n→∞ xn= sup {xn|n∈
N
}.
b) If (xn)nis decreasing, then lim
n→∞ xn= inf {xn|n∈
N
}.
2) Let (xn)nbe a sequence of real numbers. Show the following:
a) xn→a⇔ ∀ n0∈
N
∃N0∈
N
such that |xn−a|<1
n0for all n>N0.
a) xnis Cauchy ⇔ ∀ n0∈
N
∃N0∈
N
such that |xn−xm|<1
n0for all m, n > N0.
3) Let (xn)nand (yn)nbe sequences of real numbers s.t. yn≤xnfor n>> 0. Prove the following:
a) Comparison principle: If (xn)nand (yn)nare convergent, then limyn≤lim xn.
Is the same true if one replaces “ ≤” by “ <”?
b) “Sandwich” principle: If (zn)nis a sequence s.t. yn≤zn≤xnfor n >> 0, and lim yn= lim xn, then
(zn)nis convergent, and limzn= lim xn.
4) Consider a non-empty subset X⊂
R
. Prove or disprove the following:
a) Xis bounded ⇔Every sequence (xn)nwith xn∈Xhas a Cauchy sub-sequence.
b) Xhas upper bounds ⇔Every increasing sequence (xn)nwith xn∈Xis Cauchy.
5) Cauchy sequences and real numbers. Let X⊂
R
be a subset which contains
Q
and has the property
that every Cauchy sequence (xn)nwith xn∈
Q
has a limit in X. Prove that X=
R
.
Can you give an interpretation of the above assertion?
6) Orders of growth/magnitude. Using the fact that 1
n→0, prove the following facts:
a) For a∈
R
,a > 1, and all k∈
N
one has: lim
n→∞ nk/an= 0.
b) For a∈
R
,a > 1, and all k∈
N
one has: lim
n→∞(logan)k/n = 0.
b) Show that lim
n→∞ n1
n= 1.
[Hint: Prove first that the corresponding sequences are monotone for n>> 0, etc.]
Language: One says that “exponential growth is much faster than polynomial growth”, respectively “poly-
nomial growth is much faster that logarithmic growth”.
7) Decimal representation of real numbers. In the sequel we consider decimal representations and
generalization of these as follows. Let k > 1 be a fixed natural number. A rational number a∈
Q
is called
k-ary, if its denominator is a power of k. Answer the following:
a) What are the 10-ary rational numbers? What are the 2-ary rational numbers?
b) The set Rkof all the k-ary rational numbers is closed w.r.t. addition and multiplication, hence Rkis a
subring of (
Q
,+,·). Is Rka field?
c) Explain what should be the k-ary representation of a real number. Is such a representation unique, i.e.,
is “the k-ary representation” a mathematically correct formulation?
∗8) Show that
R
is not countable, more precisely, that |
R
|=|P(
N
)|.
[Hint: Use the fact that |
N
|<|P(
N
)|, and that one has a surjection ı:P(
N
)→[0,1], N7→ Pi∈N
1
2i, etc.]