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\noindent {\bf Question}

\noindent Given a function $f:A\rightarrow {\bf R}$, define a new
function $-f:A\rightarrow {\bf R}$ by $(-f)(a) = -f(a)$.  Prove
that $\inf(-f) =-\sup(f)$.

\medskip

\noindent {\bf Answer}

\noindent Let $c =\sup(f)$, so that $c =\sup\{ f(a)\: |\: a\in
A\}$. In particular, $c\ge f(a)$ for all $a\in A$, and if $u$ is
any number satisfying $u\ge f(a)$ for all $a\in A$, then $u \le
c$. Multiplying by $-1$, we see that $-c\le -f(a)$ for all $a\in
A$ and that if $s$ is any number satisfying $s\le -f(a)$ for all
$a\in A$, then $s\ge -c$.  However, this is exactly the definition
that $-c =\inf(-f)$, as desired.


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