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1.
(a)

(b)

2.

So the completeness axiom of real number system is equivalent to the Greatest Lower Bound Property.

1.
Let A be a non-empty subset of R. Define -A := {-X I X‘kA}
Show the following statements.
(a) A has a supremum if and only if -A has an ifimum, in
which case we have inf(-A) = -supA.
(b) A has an in mum if and only if -A has a supremum, in
which case we have sup(-A) = -inf A.

2.
Show that the completeness axiom of real number system (i.e.
the Least Upper Bound Property) is equivalent to the Greatest
Lower Bound Property: Every non-empty set A of real numbers
that has a lower bound has a greatest lower bound.
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