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Preparation seminar

This seminar was given for students about to start Real Analysis II, just before Semester 2 2022.ÌýDavid 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.

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 toÌýknow what functions look likeÌýbased on their formulas
• be able to interpretÌýabsolute value statementsÌýin terms of distance (for example, reading |x-2|<3 as "the distance between x and 2 is less than 3")
• remember how to work withÌýlimits
• remember how infinite series work
• be comfortable with reading and attackingÌýproofs

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.:

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).

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Infinite series

This handout lists the tests for convergence you learned in Maths IB (you will learn even more in Real Analysis II).

• Tests for convergence handoutÌý

This seminar was given inÌý2015 for students in Maths IB. It coveredÌýmost of the topic of infinite series, including tests for convergence and intervals of convergence.

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.

This seminarÌýis about attacking proofs in a calculus context. Some examples of proofs about continuity and differentiability were given.

This seminar was given for Maths IM in 2015 and it covered logic and proofs.Ìý

The first part covered constructions such as negation, the converse and the contrapositive. The second part covered mathematical induction.

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.

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.

2021

This revision seminar was given in Semester 2 2021.ÌýDavid discussed countability of sets, and also series convergence tests (starting at 40m39s).

• (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.

• (No notes available for 2020 seminar)

2019

This revision seminar was given in Semester 2 2019.ÌýDavid 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).

• (NoÌý notes available for 2019 seminar)

2017

This revision seminar was given in Semester 2 2017. DavidÌýtook 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

2016

This revision seminar was given to students ofÌýReal 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.)

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!

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