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Example Question #5 : 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 .
Possible Answers:
True
False
Correct answer:
True
Explanation:
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,
exists and
.
Therefore proving the statement in question true.
Donald
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