Supremum of $\mathbb N$ There is no upper bound of the set of natural numbers $\mathbb N$, and so it is said that the supremum of  $\mathbb N$ is $\infty$. But the $\infty \not\in \mathbb R$, then why don't we say that supremum of $\mathbb N$ does not exist?
 A: The answer depends on the definition of supremum, and the underlying context. I will use the following definition, paraphrased from Rudin's Principles of Mathematical Analysis:

Let $S$ be an ordered set, and $E$ be a subset of $S$ which is bounded above. Suppose there is an $\alpha\in S$ with the following properties:
(i) $\alpha$ is an upper bound of $E$.
(ii) If $\beta<\alpha$, then $\beta$ is not an upper bound of $E$.
Then, the supremum of $E$ is defined as $\alpha$.

If $S=\Bbb{R}$ with the usual ordering of the reals, then because $\Bbb{N}$ is not bounded above in $\Bbb{R}$, $\sup\Bbb{N}$ does not exist. According to the above definition, it is plainly incorrect to write $\sup\Bbb{N}=\infty$, as $\infty\not\in\Bbb{R}$.
However, if $S$ is the extended real number system $\overline{\Bbb{R}}$, i.e. the real numbers along with two other elements $\infty$ and $-\infty$, then things are very different. We extend the usual order of $\Bbb{R}$ to $\overline{\Bbb{R}}$ by defining $\infty$ as the greatest extended real number. Since $\infty$ is an upper bound of $\overline{\Bbb{R}}$, it is an upper bound of $\Bbb{N}$. Moreover, it is the least upper bound as no element of $\Bbb{R}\cup\{-\infty\}$ is an upper bound of $\Bbb{N}$. Therefore, in this context, it is perfectly correct to write $\sup\Bbb{N}=\infty$.
Here is where things get tricky. Some people are happy to write $\sup\Bbb{N}=\infty$ even if $S=\Bbb{R}$. In this context, saying that $\sup\Bbb{N}=\infty$ is nothing more than a shorthand for saying that $\Bbb{N}$ has no upper bound in $\Bbb{R}$. It does not mean that there is a real number $\infty$ which is the supremum of $\Bbb{N}$. For obvious reasons, I find this convention misleading, but it is undoubtedly convenient.
A: At a fundamental level, $\sup$ is really a partial (class?-)function of two variables: an "ambient" ordered set $(X,\le)$ and a subset $S\subseteq X$.
Now, usually the ambient space is clear from context, but in the case of numerical sets authors may have a different opinion on what $(X,\le)$ should be. For instance, I often like the idea that, when I consider $\sup$ and $\inf$ of subsets of $\Bbb R$, I'm actually working with $X=[-\infty,\infty]$, and therefore I would generally write $\sup[0,\infty)=\infty$, $\inf\Bbb Z=-\infty$ and $\sup\emptyset=-\infty$. To be fair, I only say that a least upper bound or greatest lower bound exists when it is a real number. A couple of times, when I was doing stuff which inherently was about non-negative real numbers, I realized that I would have had a better time working with $X=[0,\infty]$ (because it was convenient to have $\sup\emptyset=0$ instead of $-\infty$).
Other people like to work with $X=\Bbb R$ and use a convention that is symbolically equivalent, while giving it the meaning that $\sup S=\infty$ is a standalone notation that indicates a subset unbounded above; they may or may not refuse to consider the least upper bound of $\emptyset$. Any of that works just as fine and it's a very reasonable and common point of view. Notice that, in the instance of $X=\Bbb R$, there is virtually no need to distinguish the case of a non-existent least upper bound from the case of a subset that isn't bounded above, because of $\Bbb R$ being order-complete
A: Saying the supremum is $\infty$ is the same as saying it does not exist because there is no upper bound. It's just a matter of notation.
A: If $A \ne \emptyset$ is a subset of $ \mathbb R$ such that $A$ is not bounded from above , it is common practice to write
$$\sup A = \infty.$$
A: You are correct.
The definition of a supremum of a real set $\ A\subseteq\mathbb{R}\ $ is the real number - if it exists - that is the least upper bound of $\ A.$ If no such real number exists, then no supremum of $\ A\ $ exists.
If a set is not bounded above, then clearly no upper bound exists and so no least upper bound exists, i.e. no supremum exists.
All that said, you know how in limits we sometimes write $\ \lim_{n\to\infty} a_n = \infty.\ $ This notation can be considered an "abuse of notation", but it is clear and unambiguous what this means: that given $\ r\in\mathbb{R},\ $ there exists $\ N\in\mathbb{N}\ $ such that $\ a_n\geq r\ $ for all $\ n\geq N.$ Here I am demonstrating that "abuse of notation" in maths is commonplace, and some people think that abuse of notation is fine, so long it is clear what is meant by it. Other people don't think abuse of notation is fine in mathematics. It's sort of a personal opinion.
When we say $\ \sup(A) = \infty,\ $ what we mean is that no supremum exists. So whilst on the one hand the fact that $\ \infty\ $ is not a real number suggests that the phrase $\ \sup(A) = \infty\ $ can not have any meaning, in fact it is an "abuse of notation" and means something that to me seems unambiguous: that $\ \sup(A)\ $ is not a real number, and the only way for a set of real numbers to not have a real number as it's supremum, is for the set to not be bounded above. So that's what $\ \sup(A) = \infty,\ $ means: that the set of real numbers $\ A\ $ is not bounded above.
