1. ## Bounds

1. Let $A$ be a nonempty set of real numbers which is bounded below. Let $-A$ be the set of numbers $-x$, where $x \in A$. Prove that $\inf(A) = -\sup(-A)$.

Intuitively this makes sense if you draw it on a number line. But I am not sure how to formally prove it.

2. does it just tautologically follow from the definitions?

3. We know that A has a greatest lower bound, $k = \inf (A)$.
Thus, $x \in A\quad \Rightarrow \quad k \leqslant x\quad \Rightarrow \quad - x \leqslant - k$.
That means that $-k$ is an upper bound for $-A$ so let $j = \sup ( - A)$.
We know that $j \leqslant - k$ so suppose that $j < - k$.
Then $k < - j\quad \Rightarrow \quad \left( {\exists t \in A} \right)\left[ {k \leqslant t < - j} \right]\,or\,j < - t \leqslant - k$