Class on 29-09-2026

Because there was no lecture this week, we did some more exercises from the topics already on the menu (graphing functions, dealing with bounds/infima, getting familiar with notation).

There was a blackboard, so I didn't write notes on the tablet during class. Our basic agenda was:

• Investigating the supremum/infimum of $\{ \frac{n}{n + 1} \}$

• Contemplating the idea of relations between sets

• Graphing the relation $\lvert x \rvert - \rvert y \rvert = 1$, determining whether it is a function

• Individual work on 1b, 2a, and (if you wanted a challenge) 1c from the exercise sheet

• Review of problem 1b

After the class I wrote the notes below. If you have questions or want anything else included you can email me at christopher.gadzinski@uni.lu.

Problem 1b

What is the infimum/supremum of $\{ x - y \mid x, y \in \R, \; 1 < x < 2, \; 3 < y < 4\}$?

There's no particular "formula" to lead us to the answer. My personal strategy would be to realize that this set must be an open interval, and then ask how I can make $x - y$ as large as possible/as small as possible subject to the constraints on $x$ and $y$. (To maximize $x - y$ I want $x$ to be large and $y$ to be small, so I send $x$ to its upper bound and $y$ to its lower bound, and so on.) I explained a little more about how we can approach this problem in class.

Geometric approach

Analytic approach

(Correction: I mean that $-1 - \varepsilon \in A$ for all sufficiently small positive $\varepsilon$, of course.)

Problem 1c

What is the infimum/supremum of $C = \{ \sin(k) \mid k \in \N \}$?

This is a challenging problem. We won't include a problem this hard on an exam. I included it as a curiosity, where (if you like) you can try to apply more sophisticated supremum/infimum reasoning.

What's happening in this problem? You could picture a point $(\cos(k), \sin(k))$ moving around on a circle in steps of $1$ radian. Let's think about the set of possible angles our point will make with the $x$ axis, namely $$D = \{ 2 \pi r + k \mid r \in \Z, k \in \N \} \cap [0, 2 \pi).$$ I claim that $D$ has positive elements that are arbitrarily close to $0.$ From this it follows that our point makes an arbitrarily close approach to any point on the circle, and in particular its $y$ coordinate will come arbitrarily close to the extreme values of $-1$ and $1.$

Can you prove that $D$ has arbitrarily small positive elements? (Hint: suppose that the infimum of $D \setminus \{0 \}$ were positive. What would it mean if this set didn't contain its infimum? What would it mean if it did?)