MAT2400 - Real Analysis

Background material

The material on this page summarises some concepts we will use frequently in this course.

How to work through this material

Each video corresponds (roughly) to a section in Chapter 1 and 2 in Spaces. You should:

  • Watch the video (or just skim it, if you think you already know it)
  • Read (or skim) the corresponding section in Spaces
  • Do as many of the accompanying exercises as you can.

If you're stuck, don't worry. Rewatch the video, reread the section in Spaces, reread earlier sections, discuss with a fellow student, or simply skip the exercise. You can ask questions to the group teacher during the tutorials, or ask questions in this padlet.

When you are certain that you have completed an exercise (or you have given up completely), you can check the solution (not all exercises have worked solutions, unfortunately).

The more exercises you do, the better! If you should run out of exercises, don't be afraid of doing exercises which aren't listed here.

Material

Video Section in Spaces Exercises

1. Mathematical logic (PDF)

 1.1

1.1.1–1.1.3 (in 1.1.1, prove the statement both by reductio ad absurdum and by proving the contrapositive)

2. Sets (PDF)

 1.2
  • 1.2.1–1.2.4
  • Prove Proposition 1.2.1 (have a peek at the proof in the book if you need a hint)

3. Computing with sets (PDF)

 1.2, 1.3
  • 1.2.8
  • 1.3.1–1.3.6

4. Functions (PDF)

 1.4
  • 1.4.1–1.4.8 (in 1.4.7a it should read \(f(f^{-1}(B)) \subseteq B\))
  • Try to prove Propositions 1.4.1–1.4.4

5. Cardinality (PDF)

(On p. 5, \(g\circ f\) should have been \(f\circ g\).)

 1.6

1.6.1–1.6.4. (Hint for 1.6.4: Show first the following claim: There exist bijective functions \(g:\mathbb{N}\to A\) and \(h:\mathbb{N}\to B\).)
6. Vector spaces and norms (PDF) 5.1 5.1.1–5.1.5 (In 5.1.2, replace "compact metric space" by "the interval [0,1]")
7. Convergence and continuity (PDF) 2.1
  • Let \(p>0\) be a given number and define the sequence \(x_n = n^{-p}\). Show that \(x_n \to 0\) as \(n \to \infty\).
  • ????Show that the functions \(x \mapsto |x|\) and \(x \mapsto x^2\) (for \(x\in\mathbb{R}\)) are continuous.
  • 2.1: 1, 2, 4, 7
8. Min, max, inf and sup (PDF) 2.2

2.2.1–2.2.5
9. Completeness of \(\mathbb{R}^m\) (PDF) 2.2
  • 2.2.10
  • Show that any convergent sequence in \(\mathbb{R}^m\) is also Cauchy, and vice versa.
10. Four important theorems (PDF) 2.3 2.3: 1, 2, 4, 6, 9

 

Published Jan. 7, 2021 5:21 PM - Last modified Feb. 24, 2023 11:50 AM
E-mail this page