Exercise 1.3.6

Compute, without proofs, the suprema and infima of the folowing sets:

1. $\displaystyle\{n \in {\mathbb N}:n^2 <1\}$.
2. $\displaystyle\{n/(m + n) : m, n \in {\mathbb N}\}$.
3. $\displaystyle\{n/(2n + 1) : n \in {\mathbb N}\}$.
4. $\displaystyle\{n/m : m, n \in {\mathbb N} with m + n \leq 10\}$.

I'm going to give this one a try.

1. This set definition is only a complicated definition of the $\emptyset$ so it has neither an infimum or supremum.
2. The infinum of the set is $0$ and the supremum is $1$.
3. The infinum of the set is $\frac{1}{3}$ and the supremum is $\frac{1}{2}$.
4. The infinum of the set is $\frac{1}{9}$ and the supremum is $9$.

It agree with what jmincey wrote but have a couple of comments and a single question.

1: I know that $\mathbb{N}$ sometimes is defined as $\mathbb{N}={0,1,2,...} and for some reason I found it funny that in that case both the infinum and the supremum would be 0

4: latex recognized whitespace within a \text{} but not within \$ \$ so if we wanted the "with" to be more seperated we could use the code {n/m:m,n∈N\text{ with }m+n≤10}.

$$\{n/m:m,n∈N\text{ with }m+n≤10\}$$

Q4: I would have used {n/m:m,n∈N:m+n≤10}. Do you prefer "with" instead of " : "? Or does it not matter?

Nis - I don't think a colon is appropriate where you suggest. I definitely prefer "with", particularly your nicely typeset version.