The infinite case – "Math for Non-Geeks"

In order for a set to have a real-valued supremum, it must be nonempty and bounded above. If it is unbounded above, we assign it the improper supremum $\mathrm{\infty }$. Here, we will define what this means exactly. In addition, we will assign an improper supremum and 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. That means, every $S\in ℝ$ 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:

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

In that case, $M$ has no real upper bound. We formally assign it the improper supremum $\mathrm{\infty }$ and write

$supM=\mathrm{\infty }$

Attention! The symbol $\mathrm{\infty }$ does not define a real number. So $supM=\mathrm{\infty }$ is not a supremum of $M$. Instead, it is an improper supremum. The symbol $\mathrm{\infty }$ can be used as an element of the extended real numbers, but it is not a real number and does not obey all the ordinary rules of real-number arithmetic. For example, one may define $3+\mathrm{\infty }=\mathrm{\infty }$, but one cannot then subtract $\mathrm{\infty }$ from both sides: $\mathrm{\infty }-\mathrm{\infty }$ is undefined.

Definition (improper supremum)

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

$supM=\mathrm{\infty }$

Warning

The adjective "improper" is important. $\mathrm{\infty }$ is not a real number, and it is not a proper supremum. In case $supM=\mathrm{\infty }$ , 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 below if, for every $S\in ℝ$, there is an $x\in M$ with $x<S$. In that case, we write

$infM=-\mathrm{\infty }$

Improper supremum and infimum of the empty set

The empty set also has no real-valued supremum or infimum: We consider $M=\mathrm{\varnothing }$, then a supremum would formally be the smallest upper bound.

Question: What upper bounds does $\mathrm{\varnothing }$ have?

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

Question: What is the smallest upper bound of $\mathrm{\varnothing }$?

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

$\begin{array}{cc}sup\mathrm{\varnothing }& =-\mathrm{\infty }\\ inf\mathrm{\varnothing }& =\mathrm{\infty }\end{array}$

Again, be cautious: Neither of these is a real number! The set $\mathrm{\varnothing }$ has no supremum, but an improper supremum, instead. And the same holds for the infimum. Always keep proper and improper suprema and infima strictly apart!