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Further Topics in Analysis: Solutions 7
1. (a) Prove straight from the denition (that is, without appealing to Cantors theorem
on uniform continuity) that the function f (x) = x3 is uniformly continuous on
[0, 1].
(b) Prove th
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Further Topics in Analysis: Supplementary Exercises
1. Let F = cfw_f : (0, 1) R be the set of all functions from the segment (0, 1) to R. Prove
that card(0, 1) < card(F).
Hint: Consider the set of characteristic functions
fA (
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Further Topics in Analysis: Solutions to Supplementary Exercises
1. Let F = cfw_f : (0, 1) R be the set of all functions from the segment (0, 1) to R. Prove
that card(0, 1) < card(F).
Hint: Consider the set of characteristic f
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Mini-Revision Sheet
1. What does it mean for a set A to be countable? Is the set of all rationals with prime
denominator countable? Justify your answer.
2. What does it mean for the cardinality of a set X to be greater than or
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Further Topics in Analysis: Solution to Sample Exam Question
B. Let A R and (fn (x)nN be a sequence of functions from A to R.
(a) (6 marks)
i) What does it mean to say that the sequence (fn (x)nN converges pointwise
on A to a
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Mini-Revision Sheet: Solutions
1. What does it mean for a set A to be countable? Is the set of all rationals with prime
denominator countable? Justify your answer.
A set A is said to be countable if and only if there is a bije
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Further Topics in Analysis: Sample Exam Question
B. Let A R and (fn (x)nN be a sequence of functions from A to R.
(a) (6 marks)
i) What does it mean to say that the sequence (fn (x)nN converges pointwise
on A to a function f :
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Further Topics in Analysis: Solutions 10
1. Show that if a sequence (gn )nN of Riemann-integrable functions gn : [0, 1] R converges uniformly to a function g : [0, 1] R, then g is Riemann integrable.
Since gn g uniformly, give
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Further Topics in Analysis: Solutions 4
1. Let X and Y be sets.
(a) Prove that P(X Y ) = P(X) P(Y ).
(b) Give an example to show that P(X Y ) is not always the same as P(X) P(Y ).
(c) Give an example where X = Y and P(X Y ) =
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Further Topics in Analysis: Solutions 9
1. Let f : [0, 1] R be dened by f (x) = x. Prove that f is Riemann integrable and
compute
1
f (x) dx
0
as the limit of upper (and lower) sums.
Solution. Let Pn be the uniform partition o
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Further Topics in Analysis: Solutions 11
1. Let f : [0, 1] R be a continuous function satisfying
x
1
f (t) dt =
0
for all x [0, 1].
f (t) dt
x
Show that f (x) = 0 for all x [0, 1].
x
Dene F (x) = 0 f (t) dt for every x [0, 1].
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Further Topics in Analysis: Solutions 8
1. Find the pointwise limit of the following sequences of functions on the segment [0, 2].
Is this convergence uniform on [0, 2]?
x
(a) fn (x) = n ;
(b) fn (x) =
(c) fn (x) =
x
nx+1 ;
xn
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Further Topics in Analysis: Solutions 2
We assume that all sets in questions 2 and 3 are disjoint.
1. Prove that the following sets are countable.
a) The set cfw_2, 4, 6, 8, 10, . . . of positive even numbers.
b) The set cfw_
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Further Topics in Analysis: Solutions 6
1. Let (an )nN be a sequence in (0, ). Set
l := lim sup a1/n .
n
n
(i) Show that if 0
n=1 an
l < 1, then
n=1 an
(ii) Show that if l > 1, then
converges.
diverges.
(iii) Find [0, ) for wh
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Further Topics in Analysis: Solutions 3
1. Prove that the set of all polynomials with rational coecients is countable. [Hint: First,
prove that the set of polynomials of degree n is countable. Then use Lemma 3.18.]
Solution. T
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Further Topics in Analysis: Solutions 5
1. Let (an )nN be any sequence of real numbers. Which of the following must be true?
Give counterexamples for those that may be false.
(a) (an )nN has a convergent subsequence.
(b) If (a
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Further Topics in Analysis: Exercises 8
1. Find the pointwise limit of the following sequences of functions on the segment [0, 2].
Is this convergence uniform on [0, 2]?
x
(a) fn (x) = n ;
(b) fn (x) =
x
nx+1 ;
(c) fn (x) =
xn
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Further Topics in Analysis: Exercises 9
1. Let f : [0, 1] R be dened by f (x) = x. Prove that f is Riemann integrable and
compute
1
f (x) dx
0
as the limit of upper (and lower) sums.
2. Prove that the function f (x) =
x is Rie
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Further Topics in Analysis: Exercises 11
1. Let f : [0, 1] R be a continuous function satisfying
1
x
0
for all x [0, 1].
f (t) dt
f (t) dt =
x
Show that f (x) = 0 for all x [0, 1].
2. Let f : R R be a continuous function, and
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Further Topics in Analysis: Exercises 10
1. Show that if a sequence (gn )nN of Riemann-integrable functions gn : [0, 1] R converges uniformly to a function g : [0, 1] R, then g is Riemann integrable.
2. Let f be a continuous r
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Further Topics in Analysis: Exercises 7
1. (a) Prove straight from the denition (that is, without appealing to Cantors theorem
on uniform continuity) that the function f (x) = x3 is uniformly continuous on
[0, 1].
(b) Prove th
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Further Topics in Analysis: Exercises 4
1. Let X and Y be sets.
(a) Prove that P(X Y ) = P(X) P(Y ).
(b) Give an example to show that P(X Y ) is not always the same as P(X) P(Y ).
(c) Give an example where X = Y and P(X Y ) =
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Further Topics in Analysis: Exercises 6
1. Let (an )nN be a sequence in (0, ). Set
l := lim sup a1/n .
n
n
(i) Show that if 0
l < 1, then
(ii) Show that if l > 1, then
n=1 an
n=1 an
converges.
diverges.
(iii) Find [0, ) for wh
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Further Topics in Analysis: Exercises 2
1. Prove that the following sets are countable.
a) The set cfw_2, 4, 6, 8, 10, . . . of positive even numbers.
b) The set cfw_2, 3, 5, 7, 11, . . . of prime numbers.
c) The set cfw_(x,
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Further Topics in Analysis: Exercises 3
1. Prove that the set of all polynomials with rational coecients is countable. [Hint: First,
prove that the set of polynomials of degree n is countable. Then use Lemma 3.18.]
2. Let a, b
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Further Topics in Analysis: Exercises 5
1. Let (an )nN be any sequence of real numbers. Which of the following must be true?
Give counterexamples for those that may be false.
(a) (an )nN has a convergent subsequence.
(b) If (a
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Further Topics in Analysis: Exercises 1
1. Find the domains and the ranges of the following relations
(a) R = cfw_(x, y) R2 | y > x2 .
(b) G = cfw_(x, y) R2 | x2 + y 2 = 1.
(c) R = cfw_(x, y) R+ R+ | x + y 1.
2. For each of th
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