Assume $ E \subset \mathbb{R}$ is nonempty and bounded above. Define $F:\{-x:x\in E\}$. Prove that F is nonempty and bounded below and that $\inf F = - \sup E$.
Here's my rough proof:
F is nonempty because $-1 \in F$. I know for bounded below we need $ \exists \beta \in E$ such that $ x \geq \beta \ \forall X \in F$.
Other than this I'm not sure how to proceed. Any help would be much appreciated!
I understand there have been similar arguments made for proving inf(A)=-sup(-A), but I'm very confused now because it seems like I need to prove inf(-A)=-sup(A)
Since $\sup E \geq x$ for every $x \in E$, then $-\sup E \leq -x$ for every $x \in E$. This proves $-\sup E$ is a lower bound of $F$, i.e. $- \sup E \leq \inf F$. I will leave the rest to you.