Math for Non-Geeks/The infinite case

{{#invoke:Math for Non-Geeks/Seite|oben}} 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 $\mathrm{\infty }$". 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 $S\in ℝ$ must not be an upper bound, so there is an element $x\in M$ with $x>S$. That's already the mathematical definition:

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

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

Math for Non-Geeks/Template:Definition

Analogously for sets unbounded from below:

Improper supremum and infimum of the empty set

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

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

Again, be cautious: This is not 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/ infima strictly apart!

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