All Introduction to Analysis Resources
Example Question #4 : The Real Number System
Determine whether the following statement is true or false:
If is a nonempty subset of , then has a finite infimum and it is an element of .
According to the Well-Ordered Principal this statement is true. The following proof illuminate its truth.
Suppose is nonempty. From there, it is known that is bounded above, by .
Therefore, by the Completeness Axiom the supremum of exists.
Furthermore, if has a supremum, then , thus in this particular case .
Thus by the Reflection Principal,
Therefore proving the statement in question true.