Let (-A). Then A. So -xinf(A). This implies that x-inf(A).
So -inf(A) is an upper bound of -A.
Now let y be any upper bound of -A.
Let xA. Then (-x)(-A). Because y is an upper bound of -A, -xy. This implies that x-y. This means that -y is a lower bound of A. Now, since inf(A) is the greatest lower bound of A, it follows that -yinf(A), which implies that y-inf(A).
In this way we have proved:
1) -inf(A) is an upper bound of -A
2) -inf(A) is smaller than any other upper bound of -A
This means, by definition, that sup(-A)= -inf(A).