Let $A \subset \mathbb{R}$ be nonempty and bounded below, and let $\lambda \in \mathbb{R}$ satisfy $\lambda < 0$. Then the set $\{\lambda a : a \in A\}$ is nonempty and bounded above, and
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\begin{align*}
\sup\{\lambda a : a \in A\} = \lambda \inf A.
\end{align*}