The infinite case – Serlo

In order for a set to have a supremum, it must be bounded from above. If it is unbounded, then there is a "formal supremum of $\infty$". Here, we will define what this means exactly. In addition, we will assign a supremum/ infimum to the empty set.

Improper suprema and infima for unbounded sets

A set $M$ is unbounded from above if it has no upper bound. Than means, every $S\in \mathbb {R}$ must not be an upper bound, so there is an element $x\in M$ with $x>S$. That's already the mathematical definition:

Definition (unboundedness from above for sets)

A set $M$ is unbounded from above if it has no upper bound:

$\forall S\in \mathbb {R} \,\exists x\in M:S<x$

In that case, the upper bound of $M$ is formally $\infty$, since for every element $x\in m$ there is $x<\infty$. We hence write

$\sup M=\infty$

Attention! The symbol $\infty$ does not define a real number. So $\sup M=\infty$ is not a supremum of $M$. Instead it is an improper supremum. Mathematicians spent some considerable effort into defining the object $\infty$ as a number. Their conclusion was that this cannot be done in a meaningful way: treating $\infty$ as a number would violate axioms of how to compute with numbers. For instance, we could try to define $3+\infty$. Intuitively, taking 3 and adding an infinite amount to it, we again get an infinite amount. So only $3+\infty =\infty$ makes sense. but subtracting $\infty$ from both sides gives us $3=0$, which is plainly wrong! If you've got some time, you can play a bit with infinities in your head, trying to treat them as numbers. The result will be a lot of contradictions like $3=0$ or $1=2$. This is certainly the best way to convince yourself not to treat $\infty$ as a real number ;)

Definition (improper supremum)

If a set $M$ is unbounded from above, we call $\infty$ the 'improper supremum of $M$ and write

$\sup M=\infty$

Warning

The adjective "improper" is important. $\infty$ is not a number and it is not a proper supremum. In case $\sup M=\infty$ , the set $M$ has no real-valued supremum, but only an improper supremum!

Analogously for sets unbounded from below:

Definition (improper infimum)

A set $M$ is unbounded from below, if for all $S\in \mathbb {R}$ there is an $x\in M$ with $x<S$ . In that case we write

$\inf M=-\infty$

Improper supremum and infimum of the empty set

The empty set does also not really have a supremum or infimum: We consider $M=\emptyset$, then a supremum would formally be the smallest upper bound.

Question: What upper bounds does $\emptyset$ have?

Consider any $S\in \mathbb {R}$. Is $S$ an upper bound? We have to check whether $x\leq S$ for all $x\in \emptyset$. But there is no $x\in \emptyset$. So we have nothing to check and the statement is always true. Hence any $S\in \mathbb {R}$ is an upper bound.

Question: What is the smallest upper bounds of $\emptyset$?

Any $S\in \mathbb {R}$ is an upper bound. So there is no smallest upper bound. Formally, we can go down to $-\infty$ with our upper bounds. So it makes sense to say that formally, $-\infty$ is the smallest upper bound. However, this is only a formal statement which has to be taken with a bit of caution!

Following the answers to the 2 questions above, it makes sense to define

Definition (improper supremum and infimum of the empty set)

For the empty set $\emptyset$ there is

${\begin{aligned}\sup \emptyset &=-\infty \\\inf \emptyset &=\infty \end{aligned}}$

Again, be cautious: This is not a real number! The set $\emptyset$ has no supremum, but an improper supremum, instead. And the same holds for the infimum. Always keep proper and improper suprema/ infima strictly apart!