Induction - Introduction to Analysis
Card 0 of 1
Question
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 
.
Determine whether the following statement is true or false:
If
Answer
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.
According to the Well-Ordered Principal this statement is true. The following proof illuminate its truth.
Suppose
Therefore, by the Completeness Axiom the supremum of
Furthermore, if
Thus by the Reflection Principal,
exists and
Therefore proving the statement in question true.
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