sets that do not have an infimum
Some examples for sets that do not have an infimum:
- •
The set (as a subset of ) does not havean infimum (nor a supremum). Intuitively this is clear, as the set isunbounded
. The (easy) formal proof is left as an exercise for the reader.
- •
A more interesting example: The set (again as a subset of ) .
Proof.
Clearly, . Assume is an infimum of . Now we usethe fact that is not rational, and therefore or.
If , choose any from the interval (this is a real interval, but as the rational numbers
are dense (http://planetmath.org/Dense) in the real numbers, every nonempty interval in contains a rational number, hence such a exists).
Then , but , hence and therefore is a lowerbound for , which is a contradiction
.
On the other hand, if , the argument
is very similar:Choose any from the interval . Then , but , hence and therefore. Thus contains an element
smaller than , which isa contradiction to the assumption
that ∎
Intuitively speaking, this example exploits the fact that doesnot have “enough elements”. More formally, as a metric spaceis not complete
(http://planetmath.org/Complete). The defined above is the real interval intersected with . as a subset of does have an infimum (namely ),but as that is not an element of , does not have aninfimum as a subset of .
This example also makes it clear that it is important to clearly state thesuperset
one is working in when using the notion of infimum or supremum.
It also illustrates that the infimum is a natural generalization
of theminimum of a set, as a set that does not have a minimum may still havean infimum (such as ).
Of course all the ideas expressed here equally apply to the supremum, as thetwo notions are completely analogous (just reverse all inequalities
).