Limits
Roughly speaking, lim an = L means that an approaches L as n grows. Here is the
n
precise denition of limit.
Denition 1 (Limit)
(a) A sequence of real numbers is a rule which assigns to each natural number n a real
number an . Well denote it a1 , a
Math 320 Problem Set 2
Due Wednesday, September 17
1. Let S1 and S2 be nonempty subsets of IR that are bounded from above.
(a) Prove that if S1 , S2 x IR x 0 , then
sup
xy
x S1 , y S2
= (sup S1 )(sup S2 )
(b) Find two nonempty subsets S1 and S2 of IR that
Math 320 Problem Set 6
Due Wednesday, October 22
1. Determine which of the following are metrics on IR.
a) d1 (x, y) = |x2 y 2 |
b) d2 (x, y) = (x y)2
c) d3 (x, y) = |x 2y|
2. Consider the following metrics on [1, 1].
d1 (x, y) = |x y|
d2 (x, y) = (x + 2)
Math 320 Problem Set 7.5
Not to be handed in
1. Let (E1 , d1 ) and (E2 , d2 ) be compact metric spaces. Prove that
E1 E2 , d3 (e1 , e2 ), (e , e ) =
1 2
d1 (e1 , e )2 + d2 (e2 , e )2
1
2
is compact.
2. Let
E = x [0, 1] x has a decimal expansion that conta
MATH 320 Problem Set 7 Solutions
1. Let (M, d) be a metric space and let pn nIN and qn nIN be two sequences in M that converge to p
and q respectively. Prove that
lim d(pn , qn ) = d(p, q)
n
Solution. By the triangle inequality
d(pn , qn ) d(pn , p) + d(p
MATH 320 Problem Set 7.5 Solutions
1. Let (E1 , d1 ) and (E2 , d2 ) be compact metric spaces. Prove that
E1 E2 , d3 (e1 , e2 ), (e , e ) =
1 2
d1 (e1 , e )2 + d2 (e2 , e )2
1
2
is compact.
Solution. Let xn = (en,1 , en,2 ) nIN be any sequence in E1 E2 . S
Math 320 Problem Set 8
Due Wednesday, November 12
1. Let A1 , A2 , A3 , be subsets of a metric space.
n
n
(a) Prove that if Bn =
Ai then Bn =
i=1
Ai then B
(b) Prove that if B =
i=1
Ai .
i=1
Ai .
i=1
(c) Show, by an example, that the inclusion in (b) can
Math 320 Problem Set 9
Due Wednesday, November 19
1
1
1. Let K = 0, 1, 1 , 3 , 1 , , n , IR. Prove directly from the denitions that
2
4
K is compact, totally bounded and has the nite subcover property.
2. (a) Prove that the union of a nite number of compa
MATH 320 Problem Set 8 Solutions
1. Let A1 , A2 , A3 , be subsets of a metric space.
n
n
(a) Prove that if Bn =
Ai then Bn =
i=1
Ai then B
(b) Prove that if B =
Ai .
i=1
i=1
Ai .
i=1
(c) Show, by an example, that the inclusion in (b) can be proper.
Solut
Math 320 Problem Set 10
Not to be handed in
1. Denote by 2 the set of all sequences =
x
xm
mIN
standard metric on 2 is
d( , ) = =
x y
x y
of real numbers that obey
m=1
(xm ym )2
x2 < . The
m
1/2
m=1
Dene, for each k IN,
e
(k)
2
by
(k)
j
e
=
1
0
if j = k
MATH 320 Problem Set 6 Solutions
1. Determine which of the following are metrics on IR.
a) d1 (x, y) = |x2 y 2 |
b) d2 (x, y) = (x y)2
c) d3 (x, y) = |x 2y|
Solution. None of them are metrics:
(a) d1 (1, 1) = 0 violating d(x, y) = 0 x = y.
(b) d2 (1, 1) =
Math 320 Problem Set 7
Due Wednesday, October 29
1. Let (M, d) be a metric space and let pn nIN and qn
that converge to p and q respectively. Prove that
nIN
be two sequences in M
lim d(pn , qn ) = d(p, q)
n
2. Suppose that, for each integer 1 i n, Ei is a
MATH 320 Problem Set 2 Solutions
1. Let S1 and S2 be nonempty subsets of IR that are bounded from above.
(a) Prove that if S1 , S2 x IR x 0 , then
sup
xy
x S1 , y S2
= (sup S1 )(sup S2 )
(b) Find two nonempty subsets S1 and S2 of IR that are bounded from
Math 320 Problem Set 3
Due Wednesday, September 24
1. Prove that the convergence of the sequence sn
Is the converse true?
nIN
implies the convergence of the sequence |sn |
nIN
.
2. Assume an nIN , bn )nIN and cn )nIN are sequences of real numbers that obe
Math 320 Problem Set 4
Due Wednesday, October 8
1. Let sn
nIN
be a sequence of real numbers that is bounded below. Prove that
lim inf sn = sup
n
inf sn
n>m
m
Notational clarication: For each m IN,
inf sn = inf
n>m
sn
This inf depends on m. If we write tm
MATH 320 Problem Set 3.5 Solutions
1. Inductively dene a sequence sn
and evaluate the limit.
nIN
2
by s1 = 3 and sn+1 = 3 sn +
4
3sn .
Prove that sn
Solution 1. First we guess the limit. Observe that if lim sn = s and sn+1 = 2 sn +
3
n
for all n, then
2
3
MATH 320 Problem Set 3 Solutions
1. Prove that the convergence of the sequence sn
Is the converse true?
nIN
implies the convergence of the sequence |sn |
nIN
.
Solution. Denote lim sn = s. Let > 0. By hypothesis there is an N such that |s sn | < for all
n
Math 320 Problem Set 5
Due Wednesday, October 15
1. Determine whether each of the following series converges absolutely, converges conditionally (i.e. converges but does not converge absolutely) or diverges.
(1)n
n2
n+1 n
1
n
c)
a)
b)
d)
n
n
n
n=1
n=1
n=
MATH 320 Problem Set 4 Solutions
1. Let sn
nIN
be a sequence of real numbers that is bounded below. Prove that
lim inf sn = sup
n
m
inf sn
n>m
Solution. Denote lim inf sn = s. Because the sequence is bounded below, s = . If s IR, it suces
n
to prove that,
MATH 320 Problem Set 5 Solutions
1. Determine whether each of the following series converges absolutely, converges conditionally (i.e. converges but does not converge absolutely) or diverges.
(1)n
n2
n
1
n+1 n
a)
b)
d)
c)
n
n
n
n=1
n=1 n + 1
n=2 (log10 n
MATH 320 Problem Set 9 Solutions
1
1
1. Let K = 0, 1, 1 , 3 , 1 , , n , IR. Prove directly from the denitions that K is compact,
2
4
totally bounded and has the nite subcover property.
Solution. K compact: Let cfw_an nIN be a sequence in K.
Either there i
MATH 320 Problem Set 10 Solutions
1. Denote by 2 the set of all sequences =
x
xm
mIN
standard metric on 2 is
d( , ) = =
x y
x y
Dene, for each k IN,
e
(k)
of real numbers that obey
m=1
(xm ym )2
x2 < . The
m
1/2
m=1
2
by
(k)
j
e
1
0
=
if j = k
if j = k
Math 320 Problem Set 11
Not to be handed in
1. (a) Prove that the functions f (x) = 1 and g(x) = x are dierentiable with derivatives f (x) = 0 and
g (x) = 1.
(b) Find (with justication) a function that is not continuous at any x0 = 0 but that is dierentia
Q Does Not Obey the Least Upper Bound Axiom
Recall that a eld IF obeys the least upper bound axiom if every subset E IF, that
is nonempty and bounded above, has a least upper bound in IF. In these notes we prove
that the set of rational numbers, Q, does n
Taylors Theorem
Theorem 1 (Taylors Theorem) Let a < b, n IN cfw_0, and f : [a, b] IR. Assume
that f (n) exists and is continuous on [a, b] and f (n+1) exists on (a, b). Let [a, b] and
dene the Taylor polynomial of degree n with expansion point to be
n
1 (
Math 320
Solutions to Assignment #4
October 9, 2015
Total marks = [30].
1. Draw the picture.
[2]
(a) The line in R3 through the north pole and the point (, , ) is given by r(t) = (0, 0, 1)+
t(, , 1) = (t, t, 1 + t( 1). It intersects the plane when t = ( 1
Math 320
Solutions to Assignment #3
October 2, 2015
Total marks = [30].
[2]
1. (a) Let = cfw_p Q : p < q. Then q 1 and q + 1 , so is a proper subset of Q.
1
If p and r < p then r < p < q so r , and also 2 (p + q) is rational, larger than
p, and less than
Math 320
Solutions to Assignment #1
September 18, 2015
Total marks = [30].
[5]
1. We use the following fact: if a prime integer divides a product of integers then it divides at
least one of the factors.
Proof by contradiction: suppose there are integers p
Math 320
Solutions to Assignment #2
September 25, 2015
Total marks = [30].
[1]
1. (a) Factoring, we obtain
bn 1 = (b 1) bn1 + bn2 + + b + 1 .
There are n terms in the second bracket and each is at least one, so the second bracket
is at least n, as desired
September 25, 2015
Math 320 Assignment 3: Due Friday, October 2 at start of class
1. This problem concerns Dedekind cuts, dened in the Appendix to Chapter 1.
(a) Given q Q, prove that cfw_p Q : p < q is a cut.
(b) Prove that cfw_p Q : p2 < 2 is not a cut.