Completeness Axiom

Let be a non empty subset of .

• is the upper bound of if:
• is bounded above if has an upper bound
• Maximum element of : if and is an upper bound of
• Supremum of , is the smallest upper bound of
• Maximum is a supremum. Supremum is not necessarily a maximum.
• is the lower bound of if:
• is bounded below if has a lower bound
• Minimum element of : if and is a lower bound of
• Infimum of , is the largest lower bound of
• Minimum is a infimum. Infimum is not necessarily a minimum.

Theorems

Let be a non empty subset of .

• Say is an upper bound of . Then iff:
• Say is a lower bound of . Then iff:

Proof Hint

Prove the contrapositive. Use for supremum proof.

Required proofs

Completeness property

A set is said to have the completeness property iff every non-empty subset of :

• Which is bounded below has a infimum in
• Which is bounded above has a supremum in

Both have the completeness property. doesn’t.

In addition to that:

• Every non empty subset of which is bounded above has a maximum
• Every non empty subset of which is bounded below has a minimum