Introduction to Analysis : Induction

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Example Questions

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 .

Possible Answers:

False

True

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.

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