1. analysis : infimum/supremum proof

Let A be a nonempty subset of real numbers which is bounded below. Let -A={-x|x is an element of A). Prove that: inf(A)=-sup(-A).

Any help would be greatly appreciated. Thank you.

decohen

2. Let $x\epsilon$(-A). Then $(-x)\epsilon$A. So -x$\geq$inf(A). This implies that x$\leq$-inf(A).
So -inf(A) is an upper bound of -A.

Now let y be any upper bound of -A.
Let x$\epsilon$A. Then (-x)$\epsilon$(-A). Because y is an upper bound of -A, -x$\leq$y. This implies that x$\geq$-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 -y$\leq$inf(A), which implies that y$\geq$-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).

3. If $\displaystyle \lambda = \inf (A)$ then $\displaystyle \left( {\forall x \in - A} \right)\left[ {\; - x \in A \Rightarrow \lambda \; \leqslant - x\; \Rightarrow \; - \lambda \geqslant x} \right]$.
That means that $\displaystyle -A$ is bounded above so $\displaystyle \left( {\exists \delta } \right)\left[ {\delta = \sup ( - A)} \right]\;\& \;\delta \leqslant - \lambda$.

Now suppose that $\displaystyle \delta < - \lambda$ then it follows that $\displaystyle - \delta > \lambda \, \Rightarrow \,\left( {\exists b \in A} \right)\left[ { - \delta > b \geqslant \lambda } \right]$.
BUT this means that $\displaystyle \delta < - b\;\& \, - b \in - A$ : CONTRADICTION.

4. Originally Posted by decohen@purdue.edu
Let A be a nonempty subset of real numbers which is bounded below. Let -A={-x|x is an element of A). Prove that: inf(A)=-sup(-A).

Any help would be greatly appreciated. Thank you.

decohen
Let inf(A)=u.

Let $\displaystyle y\in(-A)\Longrightarrow -y\in A\Longrightarrow u\leq -y\Longrightarrow y\leq -u$====> -A is bounded from above by -u.

So if sup(-A)=v$\displaystyle \Longrightarrow v\leq -u\Longrightarrow u\leq -v$......................(1)

Let $\displaystyle - y\in A\Longrightarrow y\in (-A)\Longrightarrow y\leq v\Longrightarrow -v\leq -y$ ====> -v is a lower bound of A.

And since inf(A)=u $\displaystyle \Longrightarrow -v\leq u$...............(2)

Combining (1) and (2) we get u=-v or inf(A)=-sup(-A)