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 ". 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[Bearbeiten]

A set is unbounded from above if it has no upper bound. Than means, every must not be an upper bound, so there is an element with . That's already the mathematical definition:

Definition (unboundedness from above for sets)

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

In that case, the upper bound of is formally , since for every element there is . We hence write

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


Definition (improper supremum)

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

Warning

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

Analogously for sets unbounded from below:

Definition (improper infimum)

A set is unbounded from below, if for all there is an with . In that case we write

Improper supremum and infimum of the empty set[Bearbeiten]

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

Question: What upper bounds does have?

Consider any . Is an upper bound? We have to check whether for all . But there is no . So we have nothing to check and the statement is always true. Hence any is an upper bound.

Question: What is the smallest upper bounds of ?

Any is an upper bound. So there is no smallest upper bound. Formally, we can go down to with our upper bounds. So it makes sense to say that formally, 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 there is

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