MAST20026: Real Analysis

5 Assignments, worth 4% each, Exam, worth 80%

Overall, I kinda felt this subject was a bit boring, and I didn't find Paul to be the best lecturer; a bit too reserved and often pulled things from nowhere in his proofs without bothering to explain them. It wasn't a terrible subject, and I found it got more interesting after Calculus (Differentiation, Riemann Integration, Series and Taylor Series) was introduced, but I didn't particularly enjoy this subject and hence engaged poorly with it, and so my marks also suffered for it.

I must admit that I did practically none of the Problem Book questions for this subject, which was probably a mistake, but I just wasn't particularly engaged in this subject. Unlike other maths subjects, there are two tutorials each week, but I honestly didn't find them to be particularly useful and hence skipped a number of them towards the end of the year.

IMO there was way too much time spent on basic Mathematical Logic (~first 4 weeks), where that time could've been better spent on later parts of the subject or simply fleshing things out better. For example, there was a ridiculously inordinate amount of time spent on truth tables, yet Taylor series were extremely rushed and crammed in at the end.

Assignments often had a mix of some very easy proof questions, and some very hard questions. The tutors generally gave marks for trying to sketch a proof though, even if not completely correct. The exam this year was a lot more 'creative' in the style of its questions, probably to compensate for COVID and online exams. There must've been some scaling of this subject to compensate for the exam difficulty imo.
Also note that, from next year, there will be a new Advanced version of this subject on offer.

Paul Norbury

Yes, 5 with solutions

3/5

None required, but I found Understanding Analysis by Abbott helpful

3x1hr Lectures, 2x1hr tutorial

2020 Semester 2

80 H1

6 written assignments due every fortnight during semester contributed to total 20%, a 3-hour written examination (80%).

The six fortnightly assignments are not particularly difficult, they are rather focused on checking if you could write a rigorous mathematical proof with all the necessary reasonings and justifications and correct two-column format, just put some time in it and check really carefully. The last assignment has an essay question which will reviewed by Deborah and it's actually interesting, let you think about how your understanding about math change through this semester.

There's really not much to say about the final exam, it has quite a routine style, much like the exams from previous years, there will be one or two tricky sub-questions but the most parts's difficulty is consistent with tutorial & exercise questions. Make sure you go through all the notes and definitions in SWOTVAC as there will be a few questions ask for writing down precise definitions and theorems which should be easy marks.


Overall I think it's a really nice second year subject to try even if you are not intending on major in math (as for math majors this is almost a must-have subject no matter which specialization). I must admit I've always considered intelligence is the most significant factor in doing math and underestimated the importance of hard-working and commitment, after this semester I wouldn't ever say so. Like Deborah said in class, this subject has the highest passing rate in all second year math subjects, if you put in effort you'll definitely get a rather satisfying result.

Coming from Calculus 2 and Linear Algebra, Real Analysis may be the most students' first brush with pure math. Unlike first year subjects and Vector Calculus which mostly concentrated on how to solve problems/calculations, it focused more on "Why" i.e. proofs behind basic mathematical theorems & precise definitions & axioms.

Yes, with screen capture.

A/Professor Deborah King

The subject starts with introducing mathematical symbols, logic operations and common quantifiers, as well as how to construct truth tables. Then we'll learn the most important part of this subject (imo), techniques of conducting proof/disproof, there are several ways, like axiomatic/direct/contrapositive/contradiction proof and proof by induction/cases. Deborah will go through each meticulously and demonstrate on a few examples, there are also a lot more similar questions on problem booklet if you're looking for more practice. After this we will methodically learn about bounds and sequences, then how to prove sequence convergence/divergence by epsilon-M definition, sequence limits and some special sequences and their proof (mainly Cauchy). Then we move to function which is basically a more "general" version of sequence (from R to R instead of from N to R), after introducing the knowledge about (deleted) neighbourhood, we again learn about function convergence and limits, then we'll goes to continuity and differentiability, learning both definitions and theorems like intermediate value theorem, mean value theorem and Rolle's theorem. Next topic is Riemann integration, Deborah will starts with Riemann sums, then refinements and finally Riemann integrable & improper integrals (all with plenty examples). The last part of this subject is series, we'll again learn about how to determine series convergence by appropriate tests, much like in Calculus 2 and we will specifically learn about power series and thus extends to using Taylor series/polynomial to approximate functions, then Fourier series as well, though we won't go too deep into these last two. There will be plenty of definitions and theorems in this subject and all of them are very important and will be examined in assignments and/or final exam, so please make sure you know them and know them well.

Deborah is a very responsible lecturer, always prepared well for class, explained definitions/theorems pretty clear and she's very keen to listen and answer questions from students both in lecture and afterwards. She also puts out a youtube channel for solving problems that were left in lectures. Though she tends to speak really fast and write excessive amount of notes at the same time (amazing ability tbh), so it's possible you cannot keep up with all the contents in class and have to watch recordings afterwards for a better understanding. And it's also normal for the lecture to go overti\me (usually 5 mins or more) so prepare for this if you got a back-to-back class at the other end of the campus.

Yes, 7 past exams from 12 to 15, some with solutions.

There are two tutorials each week which are not streamed, one on Monday/Tuesday, the other on Thursday/Friday with different tutors. Each is typical math tute, three or four people form a group and solve a question sheet on whiteboard, tutor walks between each group and answer questions. The first tutorial will focus on more theoretical side of last week's contents, while the second one will focus on practical questions of this week's contents. Since this semester's lectures were on Tue & Wed & Thur, there would be almost not enough time to watch the lectures before you went to the second tutorial if you've missed class. So better not leave a week's lectures to the weekend unless you're planning on showing up to the tutorials knowing nothing. Some tutors will also do pop quizzes about definitions before the actual tutorial starts. The solution for both tutorials' worksheets will be given at the end of the second tutorial, and the second tutor will also be the one who marks your assignments.

4.5 Out of 5

Lecture notes and problem booklet can be purchased as a bundle from co-op (10-20 bucks if I remember correctly), there's also a pdf version on lms.

3 x one hour lectures per week; 2 x one hour practice classes per week.

2016, Semester 1

H1 (82)

6 Assignments worth a combined 20% of the grade, 80% exam.

This was for sure my favorite subject I've done so far at uni. Real Analysis is an introduction to a more rigorous 'proper' style of mathematics, and is a completely different experience to calculus or linear algebra. The class starts off very gradually, you learn the rules of predicate logic and proof and apply them to very simply examples in set theory and arithmetic. It's very important you understand exactly what all of this means, because the second half of the semester will very quickly go through applications of these techniques. Seriously if you're lost in the first 3 weeks you're gonna suffer later on.

The class is all about proof and logic, very little computational maths involved. The most rigorous math subject I've done, every other subject brushes over the details of some proofs a little.

Dr. Deb was excellent. She communicates the ideas very well, has a great speaking voice, 10/10 lecturer.

However I think this is a subject that's not for everyone. Attention to detail is vital and the entire course is linked together in such a way that it's hard to get back on track if you get lost. Some classmates were frustrated at how silly and simple some of the details were.

Tutorial were a lot of fun, with thought provoking questions, and again, essentially no computational maths. They differ from first year in that students in Real Analysis tent to want be there. My tutor also had interesting insight into postgraduate maths, and opened every tutorial with open discussion about basically anything in maths.

Assignments were easy, the 6th one was an amusing essay which surprised a few people including me.

Exam had some difficult questions, but also some questions that are just free marks (truth tables!).
Subject is not for everyone, but it's exceptionally well presented.

Yes lectures are recorded and anything written by the lecturer is captured by the document camera. I never used it however.

Dr. Deborah King

Yes several recent exams, as well as older exams for past similar subjects. Worked solutions for a few exams.

5/5 For thought-provoking and well run subject.

No textbook required, the notes are comprehensive.

3x1 hour lectures per week, 1x1 hour tutorial per week.

2015 Semester 1

H2A

6 Assignments (20%) and a 3 hour end of year exam (80%)

I know there's already like three reviews for this subject so I'll try to keep it brief. The lecturer seems to change every semester and so does the emphasis of the course. If I was to sum up the course in one word it would be: epsilon. We even had to write an essay (yes you heard it right) on epsilon for the last assignment. Basically most of the main concepts are defined using epsilon proofs. You begin with Logic and Proof where you learn how to write all the basic proofs- Proofs by Induction/Contradiction/Contrapositive, Predicate logic proofs, Axiomatic proofs etc. This is arguably the most important part of the course and a solid foundation will make life easier in the topics that follow.

Then you move on to sequences where you learn to prove the convergence of sequences using Epsilon-M(or N) proofs. This part is pretty mechanical however proving limit laws are not. At first, all the different kind of proofs appear daunting but then after some practice, you start to realise that all you need to know are the definitions. Just basically:
-Figure out what you want to prove (the 'claim')
- Write down everything that is given (the 'premises')
Then use all the various definitions and theorems to get from A to B.

After sequences you learn to prove limits of continuous functions. Here, epsilon-M proofs are replaced by epsilon-delta proofs. These are slightly more complicated than sequences but mastering both is essential since they come up on exams every year. Midway into the course is IMO the more easy part of the subject. Again, definition is key here. What does it mean for a function to be Continuous? Differentiable? Integrable? You need to know how to answer these because they are your starting point for proofs and also theory questions come up in the exam too.

The last topic is series. This is the most mechanical part of the course. Basically the only advise I can give here is: practice. Do lots of series questions on the problem sheet and tutorial sheet. You will need to be careful before using the various convergence tests (comparison test, ratio test integral test, etc.) by first seeing if the series meets all of the conditions. Usually this will be "positive and decreasing". The last few lectures are on Taylor Series and Fourier series which have some important applications in areas like engineering and physics.
All in all a very well organised subject and my favourite to date.

Yes, with screen capture.

Deborah King - A good Lecturer, explains things clearly with lots of examples.

Yes. 3 given during exam period, initially with no solutions until a few days before exams (I think due to requests from students).

5 Out of 5

There is no prescribed textbook for this subject but I think there is a reading list in the lecture notes if you're enthusiastic.

3 x 1 hour lecture per week and 2 x 1 hour tutorial per week.

2013 Semester 2

96 (H1)

Ten assignments worth 20% and One 3-hour exam worth 80%

Basically, this is an introductory subject to what some might call "proper math" (for non accelerated pathway students). If you thought Linear Algebra was pedantic, then this subject will take it to a new level.

The main idea of the subject is to discuss concepts that you will be very familiar with (eg limits, differentiability, integral calculus), but discuss them in a much more rigourous way than before. The other overriding theme of the subject is PROOFS. You will learn various techniques for proving (usually simple) claims such as proof by contradiction, proof by induction, e-N and e-delta proofs. Just looking through the exam, about half of the questions are "prove that..." type questions. The subject is structured quite well in the respect that you will spend a lot of time in lectures and tutorials covering proof techniques so if you're willing to put the effort in then you won't find the proofs aspect of the subject so daunting. Having said that, if the rigourous proof-based aspect of mathematics doesn't appeal to you then you should probably think twice before taking this subject (unless you need it for something like physics). There are some simple mechanical concepts covered such as evaluating if a series converges or diverges and basic predicate logic using truth tables, but really most of the subject is proof-based. We were supposed to cover Fourier Series but disappointingly it got condensed to one haphazard lecture at the end of semester.

I found the tutorials were a good way of building knowledge in this subject. Often the lectures flew right over my head and it wasn't til the tute that the concepts started to stick. This subject is unique in the sense that there are two tutes a week, which personally i thought was really helpful.

Assignments were weekly from the second week, and there were usually 1-2 questions. They might look hard when you first get them, but spending some time thinking about them will usually result in you being able to nut out the main idea. And even if you can't then it doesn't really matter as I found the tutors were quite lenient in marking them (i got 10/20 for one assignment where the whole idea of my proof was COMPLETELY on the wrong track).

There wasn't a question book handed out like the first year subjects, but there was a problem sheet for each topic of the course. Annoyingly the lecturer left out many answers on the answers handout.

Lecturer is knowledgeable, but has near-illegible handwriting and often messes up when writing things out. You probably won't have him anyway as it seems to change each semester. It should be noted that as the lecturer in this subject changes, so does the content. Eg this semester we covered predicate logic which hadn't been done before in this subject.
Personally, I loved the content and thought the subject was organised quite well. But it isn't for everyone.

Not in my semester, but it varies from semester to semester depending upon the lecturer (there is a link to 2011 lectures in this subject here)

Dr Richard Brak

Many are available, lecturer posted answers to both 2011 exams. It should be noted that course content changes a lot from lecturer to lecturer so older exams may have many irrelevant questions

5/5

There was no prescribed textbook, a few were recommended:

• S.R. Lay, Analysis with an Introduction to Proof, 4th ed., 2004
• J. Stewart, Concepts and Contexts

Three 1-hour lectures and two 1-hour tutorials per week

2012 Semester One

H1 [84]

Ten to twelve written assignments due at weekly intervals during semester amounting to a total of up to 50 pages (20%), and a 3-hour written examination in the examination period (80%). (no multiple choice)

As stated by someone else on this thread, this subject goes into the deeper and more fundamental levels of mathematics. It's very rigorous. It's also way harder than calculus 2 and linear algebra, so I wouldn't recommend it as breadth or an elective if you found those difficult.

I found the depth that this subject went into was quite interesting, although like the other reviewer of this subject, I was frustrated with having to explain almost every minor detail in my working. For instance, you can't just use L'Hopital's rule by saying (0/0) or something without saying f is continous, g is continous bla bla bla, and can't just say the limit of 1/x as x approaches infinity is 0 without further explanation.

The labs were not assessed, but somewhat useful.

Yes, with screen capture, however, the lecturer does a lot of working out on the whiteboard which is not recorded.

Barry Hughes

Yes, about four, but as someone else on is thread pointed out, only one for this subject was useful because for some reason the lecturers all seem to teach different stuff for this one subject. The lecturer gave us plenty of practice from Accelerated Math 2 past exams though.

5 Out of 5 (in terms of interest if you like math, but don't expect it to be easy!)

None prescribed

Contact Hours: 3 x one hour lectures per week, 1 x one hour practice class per week, 4 x one-hour computer laboratory classes during semester Total Time Commitment: Estimated total time commitment of 120 hours

2011 Semester 1

H1 (barely)

12 written assignments worth 20% altogether, 80% exam.

I've come across a lot of high school students who say that they "love maths". Perfect. Then this is the subject for you. This subject will truly separate those who:

1. Truly love maths for its art & beauty
2. Appreciate maths & consider majoring in it
3. Loathe maths for its proofs, logic and deduction

It is the first subject to introduce pure maths to undergraduates for those who went through the Calculus 1,2 pathway. It is painstakingly annoying. You leave out something and/or don't consider all cases - then you can safely assume that you will be penalised. It will screw with your mind, motivation and confidence. That is a fact. It is the nature of the subject and you shouldn't expect it to be "easy". The lectures were confusing and hard to grasp the concepts. I have lost count of how many times I have said "WTF?!" in my mind. Very often you can walk out of a lecture not understanding what the hell went on in that lecture. And don't worry or fear, you won't be the only one.

This is why you need to go to the tutes. If you can walk, then rock up - even if you are dead drunk. Tutes operate a lot differently from other maths subjects. Instead of just purely working (pun not intended) on practice problems, the tutor actually goes through the important material in the lectures from the previous week and actually teaches. Then if time permits, then they'll get you to do some problems. That is why it is imperative to attend.

Despite the academic rigour of the subject and the difficulty of some of the assignments, the exam that I sat was fair. It was definitely doable. It was designed to separate those who had a thorough understanding of the course and those who just "skimmed" the surface.

Overall though, it was definitely challenging and is certainly an enriching subject to complete.


^*In 2010 semester 1, all but one exam was useless. As almost every semester there was a change in lecturer, there was a change in what was emphasised more in the course. A sample exam was given generously by Dr. Alex Ghitza, the lecturer in 2009 semester 2. Other past papers from subjects that Barry taught were also given.

Yes - except for lectures 33 & 34 (last two lectures)

Barry Hughes

Yes^*

3.5 Out of 5

None. Most of them are either beyond the course or too advanced. But Barry's top recommendation is worth a read.

3 x 1 hour lectures per week, 1 x 1 hour practice class per week, 4 x 1 hour computer laboratory classes during semester

P