MA30252: Advanced Real Analysis

The exam will not ask you to do fancy cardinality things, e.g. show that two sets have the same cardinality by cooking up a complicated bijection. But some of the facts in this section, most of which are hopefully reasonably intuitive, may be useful on the exam and indeed on the problem sheets. This section is largely an invitation to use these facts without proof.

So far the only place countability has come up is in Definition 2.5, but it will show up again in Chapter 5 and especially in Chapter 6, where one of the main hypotheses of all of the results is that we are working with a countable collection of sets (equivalently an infinite sequence of sets, see Corollary 1.68).

This is a difficult comparison to make, in part because when writing an exam I am under a great number of additional constraints which don’t apply when I’m setting the problem sheets. On the exam, for instance, there is a University/Department requirement that a certain number of the marks in each question be ‘first class marks’. Among other things, these are there to differentiate students at the extreme upper end of the mark distribution. As a result they need to be quite difficult. When writing problem sheet questions, I am completely uninterested in differentiating between students at extreme the upper end of the mark distribution. Instead, I am trying to come up with problems which will help you to learn the concepts in this unit. I would say that the hardest part of an exam question will probably be harder (in some sense) than most problem sheet questions. On the other hand, some parts of an exam question will easier than just about any problem sheet question; they will simply ask you to recall a definition or result from lectures, without testing in any way whether you understand what the definition or result means or how to use it.

Absolutely! If broken down into sub-parts guiding you through the argument, it might not even be the most difficult part of an exam question. On the other hand, without any hints whatsoever, it could potentially be one of the harder parts of an exam question.

Sometimes the official problem sheet solutions will prove one statement and then say that the proof of another statement is similar, skipping the details. You are indeed allowed to do this on the exam, of course provided that the two arguments in question really are sufficiently similar. For a relatively clear-cut example of this on the problem sheets, see for instance the solution to Exercise 2.1.1: When proving one implication the solutions establish that \(S\) is dense in \(X\text{,}\) and then say “With the same arguments we prove that \(T\) is dense in \(Y\)”.

The official line on this is in Subsection H.1.3. As an example, if you say that X implies Y by a result from lectures, and there is indeed a theorem in the lectures which says almost exactly this, then that should be completely fine. On the other hand if you what you need follows from a result from lectures only after a good ten lines of work, and superficially looks very different from what’s in the lecture notes, then then just saying ‘by a result from lectures’ may not be enough. If you’re not sure I’ll be able to follow you, you can always give me more information about the result you have in mind. For example, you could say “by a problem sheet exercise on the relationship between closures and completions”.

Yes to the first question. For example, on The 2020 exam, you would be completely free to use part (c) of Question 3 in your solution to part (d), even if you had not done part (c). The answer to the second question is often also yes. For example on the same exam, if you were able to solve part (b)(i) of Question 1, then you would also have proved part (a)(i) as a special case, and so if you made a comment to this effect you would get full marks.

On the other hand, if in this last example you gave a complete proof of (c), e.g. by showing by hand that every open cover had finite subcover, and then in (a) and (b) you simply cited your proof of (c), then this would earn full marks. The thing that matters here is whether, when taken together, your answers to (a)–(c) provide a complete proof of all three statements.

On the past exams, you will notice that the earlier parts of the question tend to be the easier ‘engagement marks’, while the later parts of the question tend to be the more difficult ‘first class marks’. While you can expect this general pattern to continue, it is not an ironclad rule. For instance, on The 2022 exam Q2(b)(ii) is arguably more difficult than Q2(c)(i).

If these rules about brackets are at all surprising or mysterious to you, then I recommend in the strongest possible terms that you carefully go through Section 1.6 in the notes and Exercise 1.6.1 on Problem Sheet 4, and then reach out to me if you have any additional questions. Confusion about notation for functions is an extremely common source of lost marks on exams in this unit.

While there have historically been more questions about results with names than about results without names, it isn’t impossible to ask about unnamed results. See for instance Question 2(b) of The 2020 exam, which asks you to “State a theorem regarding the existence of a completion …”.

Yes, certainly. I suspect we have used this fact in lectures as well.

The proof is quite fast: If \(\seq xn\) is a sequence in \(S\) converging to \(x\text{,}\) then for each \(r \gt 0\) we can find \(N \in \N\) such that \(d(x_n,x) \lt r\text{,}\) hence \(B_r(x) \cap S \ne \varnothing\text{.}\) Conversely, if \(B_r(x) \cap S\) is nonempty for all \(r \gt 0\text{,}\) then we can find a sequence \(\seq xn\) with \(x_n \in B_{1/n}(x) \cap S\) for each \(n \in \N\text{,}\) which therefore converges to \(x\text{.}\)

Certainly you can freely use any of the shorthand/abbreviations that I’ve been using in lectures. If you use other common shorthand I am usually good at guessing what you mean, but if you have very idiosyncratic shorthand you like to use it would be good to have this explained somewhere in the exam paper, with a reminder at the start of each question. (I mark question by question rather than paper by paper, and so will probably not remember what you said in Question 1 while marking your Question 3.)

They are both very useful, but in different ways. Problem sheets are examinable, and the exam is designed so at least some of them will come in handy. Because of this you will definitely want to have worked through all of the Problem sheets, even if this means ignoring some of the past exams. Past exams are not examinable (and repeating their solutions verbatim on this exam will not earn you many marks at all). On the other hand, past exams are a better guide to the style and difficulty of the exam; see one of the above questions for more on this.

Yes, and this happens a lot on past exams, and comes in a variety of difficulty levels.

Absolutely. A fair number of our proofs had some sort of recursive flavour, e.g. Theorem 4.10 and Theorem 6.1.