Exercises
The following exercises are not mandatory, but highly recommended. I deliberately give a large number of problems, and you will have to decide which to focus on.
You will find solutions to many of these problems here.
| Date | Problems |
|---|
| 19/1–25/1 | - 1.1: 1, 2, 3
- 1.2: 1–4, 8
- 1.4: 1–4, 6–8
- 1.6: 1–4
- 2.1: 1, 2, 4, 7
- 2.2: 1–5, 10, 9
- 2.3: 1, 2, 4, 9
|
| 26/1–1/2 | - ????Prove Proposition 3.1.4 (the inverse triangle inequality): If \((X,d)\) is a metric space and \(x,y,z\in X\) then \(|d(x,y)-d(y,z)|\leq d(x,z)\).
- Section 3.1: Exercises 1, 2, 6, 7
- Section 3.2: Exercises 1, 2, 5, 6, 8
- Consider \(X=\mathbb{R}\) with the canonical metric \(d(x,y)=|x-y|\). Show that the interval?\((a,b)\) is open and that \([a,b]\) is closed,?for any \(a,b\in\mathbb{R}\) with \(a\leq b\). (What happens if \(a=b\)?) Explain why?\((a,b]\) is neither open nor closed when \(a<b\).
|
| 2/2–8/2 | - Let \((X,d)\) be a metric space. Prove that every finite subset of \(X\) is closed. (A finite set is a set containing only finitely many points.)
Hint:?Show first that singletons (i.e. sets containing only one point) are closed. Next, use Proposition 3.3.13 b). - Section 3.3: Exercise 1, 2, 3, 11, 7
- Section 3.3: Let \((X,d_X)\) and \((Y,d_Y)\) be metric spaces, let \(f:X\to Y\) be a continuous function, let \(y\in Y\) be some given?point, and consider the problem of finding an \(x\in X\) such that \(f(x)=y\) (that is, we wish to solve the above equation). Prove that the set of solutions of this equation is a closed subset of \(X\).
Hint:?Phrase the question in terms of finding \(f^{-1}\) of a closed set. - Consider \(X=\mathbb{R}\) with the canonical metric \(d(x,y)=|x-y|\). The?support?of a function \(f:\mathbb{R}\to\mathbb{R}\) is the set of points \(x\) where \(f(x)\neq 0 \). Prove that the support of a continuous function is always open.
Note: In the literature, the "support" of?\(f\)?is usually defined to be the closure of the above set, which is of course closed, not open. - Section 3.4: Prove that \((\mathbb{Z},d)\), where \(d(x,y)=|x-y|\), is complete.
Hint:?What does it mean for a sequence in \((\mathbb{Z},d)\) to converge or to be Cauchy? - Section 3.4: Exercise 2, 4, 5, 6, 8
- Show that the equation?\(\cos t = 2t\)?has a unique solution.
Hint: Formulate the problem as finding the fixed point of a function?\(f\).
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| 9/2–15/2 | ? |
| 16/2–22/2 | ? |
| 23/2–1/3 | ? |
| 2/3–8/3 | ? |
| 9/3–15/3 | ? |
| 16/3–22/2 | ? |
| 23/3–29/3 | ? |
| 30/3–5/4 | ? |
| 6/4–12/4 | ? |
| 13/4–19/4 | ? |
| 20/4–26/4 | ? |
| 27/4–3/5 | ? |
| 4/5–10/5 | ? |
| 11/5–17/5 | ? |
| 18/5–24/5 | ? |
Published Jan. 15, 2026 11:14 AM
- Last modified Jan. 29, 2026 4:19 PM