# The infinite case – Serlo

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

Definition (unboundedness from above for sets)

A set ${\displaystyle M}$ is unbounded from above if it has no upper bound:

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

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

${\displaystyle \sup M=\infty }$

Attention! The symbol ${\displaystyle \infty }$ does not define a real number. So ${\displaystyle \sup M=\infty }$ is not a supremum of ${\displaystyle M}$. Instead it is an improper supremum. Mathematicians spent some considerable effort into defining the object ${\displaystyle \infty }$ as a number. Their conclusion was that this cannot be done in a meaningful way: treating ${\displaystyle \infty }$ as a number would violate axioms of how to compute with numbers. For instance, we could try to define ${\displaystyle 3+\infty }$. Intuitively, taking 3 and adding an infinite amount to it, we again get an infinite amount. So only ${\displaystyle 3+\infty =\infty }$ makes sense. but subtracting ${\displaystyle \infty }$ from both sides gives us ${\displaystyle 3=0}$, 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 ${\displaystyle 3=0}$ or ${\displaystyle 1=2}$. This is certainly the best way to convince yourself not to treat ${\displaystyle \infty }$ as a real number ;)

Definition (improper supremum)

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

${\displaystyle \sup M=\infty }$

Warning

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

Analogously for sets unbounded from below:

Definition (improper infimum)

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

${\displaystyle \inf M=-\infty }$

## Improper supremum and infimum of the empty set

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

Question: What upper bounds does ${\displaystyle \emptyset }$ have?

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

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

Any ${\displaystyle S\in \mathbb {R} }$ is an upper bound. So there is no smallest upper bound. Formally, we can go down to ${\displaystyle -\infty }$ with our upper bounds. So it makes sense to say that formally, ${\displaystyle -\infty }$ 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 ${\displaystyle \emptyset }$ there is

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

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