REAL ANALYSIS
1A
Logic and the Language of Mathematics
Important Ideas
Truth tables
Negation, conjunction, disjunction and conditional
Universal and existential quantifiers
1.
The Island of Knights and Knaves
On a mysterious island, there are two types

Real Analysis
MAST20026 Tutorial 1 Answers
1. The Island of Knights and Knaves
Ben is a Knave, and Chris is a Knight. It is impossible to decide what Alex is from the
information given.
If you want to find out what type a person is, ask a question with an

MAST20026 Real Analysis
Assignment 2, due before 2.30pm Monday 22nd August.
Please submit your assignment to your Monday/Tuesday tutors box on the ground floor of the
Richard Berry building.
Remember to include your name, tutors name and tutorial time on

Axioms for Real Numbers
We start by assuming there is a non-empty set R with two binary operations addition, and multiplication.
The two binary operations satisfy the following field axioms.
Field axioms
Equality (it is a reflexive, symmetric and transit

REAL ANALYSIS
5A
Sequences
Important Ideas
Sequences
Divergence
-M definition of convergence
Subsequences
1.
What is a Sequence?
Which of the following is not a sequence?
(a) 1, 2, 3, 4, 5, . . .
(c) 4, 9, 16, 25, 36
(b) 3, 1, 4, 1, 5, 9, . . .
(d) 1,

From now on we combine the two proofs into a single contradiction
proof .
Thus, if a theorem is in the form of a conditional:
Theorem: If P then Q
that is, we need to prove P ) Q is true, then a proof by contradiction generally
takes the form:
Proof (by c

MAST20026 Tutorial 9 Answers
1. Intermediate Value Theorem (8.8 redux)
See answers to Tutorial 8.
2. Calculating a Riemann Sum
(a) The lower Riemann sum is
L(f, P ) =
3
X
mk (xk xk1 )
k=1
= 5 2 + 3
=
3
3
+0
2
2
11
2
(b) The upper Riemann sum is
U (f, P )

MAST20026 Tutorial 2 Answers
1. The Lady or the Tiger?
Signs II and III contradict each other, so one of them is true. Since at most one sign is
true, sign I must be false. Therefore the Lady is in room I.
3. Proof by Example
(a) The roots of the equation

MAST20026 Real Analysis
Assignment Four, d ue before 2.30pm Monday 19th September
Please submit your assignment to your Monday/Tuesday tutors box on the ground floor of the
Richard Berry building (near the northern exit).
Remember to include your name, tu

REAL ANALYSIS
10A
Series
Important Ideas
Sequence of partial sums
Divergence
Convergence
Comparison Test
Absolute and Conditional Convergence
Ratio Test
Geometric series
Alternating Series Test
Harmonic p-series
Integral Test
When doing series p

MAST20026 Tutorial 4 Answers
1. Proof by Cases
Case 1:
0 x
Case Premise
|x| = x
0 < |x|
x |x|
Case 2:
Definition of |x|
x < 0
Case Premise
0 < x
|x| = x
0 < |x|
x < |x|
x |x|
Definition of |x|
Transitivity
Thus (x 0 or x > 0) = (x |x|). Since the antece

REAL ANALYSIS
2A
Sets and Simple Proofs
Important Ideas
Proof by Counterexample
Direct Proof
Sets
Subsets
Union, Intersection and Complement
Power sets
1.
The Lady or the Tiger?
In an obscure country, criminals are sentenced to Trial by Logic instea

MAST20026 Real Analysis
Assignment 3, due 2.30pm Monday 5th September
Please submit your assignment to your Monday/Tuesday tutors box on the ground floor of the
Richard Berry building (near the northern exit).
Remember to include your name, tutors name an

REAL ANALYSIS
6A
Bounded Sequences and Limit Theorems
Important Ideas
Monotonic sequence theorem
N definition of convergence
Arithmetic of Limits
1.
Pop Quiz!
2.
I am not a Proof
Here is a bad proof. What is wrong with it?
Claim: The sequence defined b

REAL ANALYSIS
8A
Functions and Differentiability
Important Ideas
Functions
Intermediate Value Theorem
Injective, Surjective and Bijective
Differentiability
Inverse of a function
Continuity
Mean Value Theorem
1.
Pop Quiz!
2.
Putting the Fun in Funct

REAL ANALYSIS
9A
Integration and Riemann Sums
Important Ideas
Riemann Sums, Lower and Upper sums
Improper Integrals of both types
Definition of the Riemann Integral
Fundamental Theorem of Calculus
Integrability
Comparison Test for improper integrals

REAL ANALYSIS
4A
Bounds, Suprema, Infima and Induction
Important Ideas
Absolute Value
Supremum
Mathematical Induction
Bounds
Infimum
Other Types of Proof
1.
Proof by Cases
Prove that x R x |x| and x |x|.
2.
The Triangle Inequality
(a) State the tria

MAST20026 Tutorial 6 Answers
3. Bounded and Monotonic Look in your notes for the definitions and theorems.
(a) The sequence an = n is bounded above by 0 but is not bounded below. The sequence
bn = (1)n is bounded by 2.
(b) The sequence (an ) in (a) above

MAST20026 Tutorial 8 Answers
2. Putting the Fun in Functions
(a) No, for example f (Australia) = f (Argentina).
(b) No, there is no country starting with X.
(c) No, since it is neither injective nor surjective.
The function can be made bijective. There ar

The University of Melbourne
Department of Mathematics and Statistics
MASTZQQQZQ REAL ANALYSlS WITH APPLECAElQNS
W Semester 2 Examination 291G
Examination Duration: 3 hours
Reading time allowed: 15 minutes
This paper has 3 pagesa including the cover sheet

Exam: 620-295 Real Analysis with applications
The University of Melbourne
Department of Mathematics and Statistics
($20-$95 Real Analysis with Applications
Examination Semester I 2010
Examination duration: 3 hours
Reading time allowed: 15 minutes
This pap

THE UNIVERSITY OF MELBOURNE
DEPARTMENT OF MATHEMATICS AND STATISTICS
SEMESTER 2, 2012
MAST20026 REAL ANALYSIS AND APPLICATIONS
Exam duration Three hours
Reading time 15 minutes
This paper has 4 pages, including this cover sheet.
Exam Papers with Common Co

THE UNIVERSITY OF MELBOURNE
DEPARTMENT OF MATHEMATICS AND STATISTICS
SEMESTER 1, 2013
MAST20026 REAL ANALYSIS
Exam duration Three hours
Reading time 15 minutes
This exam has 5 pages, including this cover sheet.
Instructions to Invigilators:
Initially, stu

THE UNIVERSITY OF MELBOURNE
DEPARTMENT OF MATHEMATICS AND STATISTICS
SEMESTER 2, 2014
MAST20026 REAL ANALYSIS
Exam duration Three hours
Reading time 15 minutes
This paper has 6 pages, including this cover sheet.
Exam Papers with Common Content:
This paper

THE UNIVERSITY OF MELBOURNE
DEPARTMENT OF MATHEMATICS AND STATISTICS
SEMESTER 1, 2014
MAST20026 REAL ANALYSIS
Exam duration e Three hours
Reading time — 15 minutes
This'exam has 5 pages, including this cover sheet.
Instructions to Invigilators:
Initially,

Q3
Maugf‘biﬁ‘aa '1,;1C)\Ur JAchL Ska aﬂcwm I
w
._.,_,_
)0) ., , 1
A, n, r ,5. WWW
‘1 P "kw W “1&9
"'1" T g: F’ T F T T
T“ 'F F F ‘T‘ 'T’ T T
F T T" T“ T F F‘ :‘i
F F iT"._ F F T” T‘ _\
Tm.— 5VDUVW~I§V 15> 0*- ‘-¥cuv~\ro\ﬂ‘3ﬂ I
q».— Mw 1m— if:

MAST20026 Tutorial 10 Answers
1. Exploration Discussion in class.
2. Divergence The theorem says that if a series
X
an converges then lim an = 0. One can use
n
n=1
the contrapositive to show that a series diverges.
1
1/n
0
= lim
=
= 0. Needs further
n n +

MAST20026 Tutorial 11 Answers
1. Radius of Convergence We first find the radius of convergence using the limit form of the ratio
test. We then check whether the series converges at the endpoints.
1
= limn |x| 1 + n1
= |x| < 1 for all |x| < 1. For x = 1, t

REAL ANALYSIS
Real Numbers, Cartesian Product, Contrapositive and Contradiction
3A
Important Ideas
Real Number Axioms
Proof by Contrapostive
Cartesian Product
Proof by Contradiction
1.
Pirate Hat Justice
The meanest pirate of them all has it in for hi