Real Analysis II
Resources for Real Analysis II - for more information about the course, please see course outlines.
-
Preparation and Assumed Knowledge
Preparation seminar
This seminar was given for students about to start Real Analysis II, just before Semester 2 2022. David discussed various concepts useful to know before doing Real Analysis II, including ways to think about functions, inequality reasoning, the need to fill in working in proofs, seeing absolute value as distance, and set notation.
- VIDEO (YouTube): Ready for Real Analysis Sem 2 2022
- VIDEO (Echo360): Ready for Real Analysis Sem 2 2022
- PDF notes from Ready for Real Analysis Sem 2 2022
Resources from earlier courses
Real Analysis is built on a lot of ideas that are in first year maths and earlier. In particular, you need to:
- be able to know what functions look like based on their formulas
- be able to interpret absolute value statements in terms of distance (for example, reading |x-2|<3 as "the distance between x and 2 is less than 3")
- remember how to work with limits
- remember how infinite series work
- be comfortable with reading and attacking proofs
These resources from other courses might help you to revise these and other skills to prepare for Real Analysis II:
Functions
This seminar was given to new international students in Summer Semester 2015, covering various different functions and their graphs, as well as the rules of differentiation.:
- VIDEO (YouTube): Functions and Differentiation seminar Summer Semester 2015
- VIDEO (Echo360): Functions and Differentiation seminar Summer Semester 2015
- PDF: Functions and Differentiation seminar notes Summer Semester 2015
Limits
This seminar was given in 2014 for students in Maths IB. It covered limits of both one and two variable functions (only the one-variable limits are relevant for Real Analysis II).
- VIDEO (YouTube): Maths IB: Limits revision seminar Sem 2 2014
- VIDEO (Echo360): Maths IB: Limits revision seminar Sem 22014
- PDF: Maths IB: Limits revision seminar notes Sem 2 2014
Infinite series
This handout lists the tests for convergence you learned in Maths IB (you will learn even more in Real Analysis II).
This seminar was given in 2015 for students in Maths IB. It covered most of the topic of infinite series, including tests for convergence and intervals of convergence.
- VIDEO (YouTube): Maths IB Revision seminar: series tests for convergence and invervals of convergence Sem 1 2015
- VIDEO (Echo360): Maths IB Revision seminar: series tests for convergence and intervals of convergence Sem 1 2015
- PDF: Maths IB Revision seminar notes: series tests for convergence and intervals of convergence Sem 1 2015
Proofs
This revision seminar is about the main theorems concerning continuous and differentiable functions: Rolle's theorem, the Intermediate Value Theorem and the Mean Value Theorem. These theorems used to be in Maths IB but you now learn them for the first time in Real Analysis II. This will be a good introduction.
- VIDEO (YouTube): Maths !B revision seminar: Calculus Theorems 2013
- VIDEO (Echo360): Maths !B revision seminar: Calculus Theorems 2013
- PDF: Maths IB revision seminar notes: Calculus Theorems 2013
This seminar is about attacking proofs in a calculus context. Some examples of proofs about continuity and differentiability were given.
- VIDEO (YouTube): Maths IB: Doing proofs in calculus 2017
- VIDEO (Echo360): Maths IB: Doing proofs in calculus 2017
- PDF: Maths IB: Doing proofs in calculus notes 2017
This seminar was given for Maths IM in 2015 and it covered logic and proofs.
The first part covered constructions such as negation, the converse and the contrapositive. The second part covered mathematical induction.
- VIDEO (YouTube): Maths IM Revision seminar: logical constructions 2015
- VIDEO (Echo360): Maths IM Revision seminar: logical constructions 2015
- PDF: Maths IM Revision seminar notes: logical constructions 2015
- VIDEO (YouTube): Revision seminar: mathematical induction 2015
- VIDEO (Echo360): Revision seminar: mathematical induction 2015
- PDF: Maths IM: Revision seminar notes: mathematical induction 2015
-
Revision seminars for Real Analysis II
The following revision seminars have been given for students in Real Anaylsis II in the past, and you might find them useful for studying the course. Note that the course has changed every semester for the last several years, so some topics or specific ideas may not be in the course this year. Even so, the ideas for attacking proofs will be useful.
2023
This revision seminar was given in Semester 2 2023. David discussed deciding how to prove things, Cauchy sequences, the fundamental theorem of calculus and the sequence definition of compact.
- Video (YouTube): Real Analysis II: Cauchy sequences and various proofs 2022
- Video (Echo360): Real Analysis II: Cauchy sequences and various proofs 2022
- PDF: Real Analysis II: Cauchy sequences and various proofs notes 2022
2022
This revision seminar was given in Semester 2 2022. Students gave various problems and we talked about the concepts related to them. We discussed uniform continuity, integration via partitions, and a little on the extreme value theorem.
- Video (YouTube): Real Analysis II: uniform continuity, integrals via partitions 2022
- Video (Echo360): Real Analysis II: uniform continuity, integrals via partitions 2022
- PDF: Real Analysis II: uniform continuity and integrals via partitions notes 2022
2021
This revision seminar was given in Semester 2 2021. David discussed countability of sets, and also series convergence tests (starting at 40m39s).
- VIDEO (YouTube): Real Analysis II: countability, series convergence 2021
- VIDEO (Echo360): Real Analysis II: countability, series convergence 2021
- (No notes available for 2021 seminar)
2020
This revision seminar was given in Semester 2 2020. David discussed the definitions of open and closed sets in R, and did a few proofs of sets being open, not open and not closed.
- VIDEO (YouTube): Real Analysis II: Open and closed sets 2020
- VIDEO (Echo360): Real Analysis II: Open and closed sets 2020
- (No notes available for 2020 seminar)
2019
This revision seminar was given in Semester 2 2019. David discussed deciding if a function is continuous and differentiable at a transition point (1m55s), and also concepts in topology such as open sets and compact sets (40m).
- VIDEO (YouTube): Real Analysis II: Continuity/differentiability and topology 2019
- VIDEO (Echo360): Real Analysis II: Continuity/differentiability and topology 2019
- (No notes available for 2019 seminar)
2017
This revision seminar was given in Semester 2 2017. David took requests from the students who came and discussed the following things, showing some examples of coming up with proofs too.
Open and closed sets: 1m50s
Uniform continuity: 23m40s
Increasing/decreasing/monotonic sequences: 49m20s
Advice for studying named theorems such as the Bolzano-Weierstraus theorem: 56m30s
Countability and infinite sets: 1h1m
Sequences of functions, including pointwise and uniform convergence: 1h24m- VIDEO (YouTube): Real Analysis II: Miscellaneous concepts 2017
- VIDEO (Echo360): Real Analysis II: Miscellaneous concepts 2017
- PDF: Real Analysis II: Miscellaneous concepts notes 2017
2016
This revision seminar was given to students of Real Analysis II in 2016. David worked through the concepts of Open and Closed, and through the meaning of the various Tests for Convergence. (The part on tests of convergence begins at about 1h5m.)
- VIDEO (YouTube): Real Analysis II: Open/closed and convergence tests 2016
- VIDEO (Echo360): Real Analysis II: Open/closed and convergence tests 2016
- PDF: Real Analysis II: Open/closed notes 2016
- PDF: Real Analysis II: Convergence tests notes 2016
2015
This revision seminar was given to students of Real Analysis II in 2015. David wrote proofs for various exam questions: the sum of a geometric series, the ratio test, absolute convergence implies convergence, and uniform convergence of a series of functions preserves continuity. He did this live without knowing the answers already so you can learn some skills for coming up with proofs on your own. Be warned that in some of them he made mistakes and had to go back and fix them so watch the whole thing!
- VIDEO (YouTube): Real Analysis II: various proofs 2015
- VIDEO (Echo360): Real Analysis II: various proofs 2015
- PDF: Real Analysis II: various proofs notes 2015
2014
This revision seminar was given to students of Real Analysis II in 2014. David wrote proofs for various theorems such as the fundamental theorems of calculus, Cauchy's MVT, the fact that a sequence's limit is unique, and proving that sequences of functions do or do not converge uniformly. He did this live without knowing the answers already so you can learn some skills for coming up with proofs on your own