A finite subset of an ordered set contains an $\inf$ and $\sup$
92249
Every finite set contains its supremum: proof improvement.
82009
Prove that every totally ordered set has a well-ordered cofinal subset
83714
$A \in B$ vs. $A \subset B$ for proofs
94825
Open set as a finite union of closed sets
76863
How to show $ \sup \inf g(x,y) \leq \inf \sup g(x,y)$?
148399
Two sets which contain each other as elements
66398
Union of a finite set and a countably infinite set is countably infinite
147941
comparing sup and inf of two sets
19414
Uncountable subset with uncountable complement, without the Axiom of Choice
12436
The cardinality of the set of all finite subsets of an infinite set
144787
Show that the set of all infinite subsets of $\mathbb{N}$ has cardinality greater than $\mathbb{N}$
22482
How can an ordered pair be expressed as a set?
136789
How to prove $a \lt b$ for all $a \in A, b \in B$ implies $\sup A \leq \inf B$?
81843
Sequences of sets property
6291
Proving something is a linearly independent subset
146098
Number of nonempty subsets of a set with $n$ elements
80180
Discrete and compact subset must be finite