Supremum & Maximum: Infinity

Question

I'm asked to tell what the supremum and maximum is of the following set: $$\begin{Bmatrix}n:n\in\mathbb{N}\end{Bmatrix}$$

I don't know that I should say $\text{sup}\begin{Bmatrix}n:n\in\mathbb{N}\end{Bmatrix}\overset{?}{=}\infty$ because that seems an abuse, but I'm not sure like I said.

Answer 1

Try writing out the first few numbers:

$$\frac{1}{2}, \frac{1}{4}, \frac 1 6, ...$$

So the supremum and the maximum agree with each other, since there's a largest element, namely $1/2$.

Edit: For clarification, the question originally asked what the supremum and maximum of $\{\frac{1}{2n} : n \in \Bbb{N}\}$ are.

Answer 2

As I and others mentioned in the comments, if the author does not have 0 in his or her definition of the natural numbers, you have a strictly decreasing sequence of rational numbers with a well-defined maximal element and supremum at 1/2. You have no well defined minimal element, but you have an infimum at 0.

Answer 3

The idea of supremum and maximum come only for a bounded set. You are considering the set $\{n : n \in \mathbb{N}\} = \{1, 2, 3, \dots\}$. This is an unbounded set in $\mathbb{R}$, as for any positive real number $G$, you shall get an element $m \in \mathbb{N}$ s.t. $m > G$.

So Supremum of the set does not exists, and same for maximum.