Let $A \subset \mathbb{R}$ be nonempty and bounded above, let $c \in \mathbb{R}$, and let $\lambda \in \mathbb{R}$ satisfy $\lambda > 0$. Then the translated set $\{a + c : a \in A\}$ and the positively scaled set $\{\lambda a : a \in A\}$ are nonempty and bounded above, and
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\begin{align*}
\sup\{a + c : a \in A\} = \sup A + c.
\end{align*}
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Moreover,
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\begin{align*}
\sup\{\lambda a : a \in A\} = \lambda \sup A.
\end{align*}