Exercise 1.3.5

Question

As in Example 1.3.7, let $A \subseteq \mathbf{R}$ be nonempty and bounded above, and let $c \in \mathbf{R}$. This time define the set $c A=\{c a: a \in A\}$.
  \begin{enumerate}[label=(\alph*)] 
  \item If $c \geq 0$, show that $\sup (c A)=c \sup A$.
  \item Postulate a similar type of statement for $\sup (c A)$ for the case $c<0$.
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] \item Assume $c > 0$ (the $c = 0$ case is trivial). Let $s = c \sup A$. Suppose $ca > s$, then $a > \sup A$ which is impossible, meaning $s$ is an upper bound on $cA$. Now suppose $s'$ is an upper bound on $cA$ and $s' < s$. Then $s'/c < s/c$ and $s'/c < \sup A$ meaning $s'/c$ cannot bound $A$, so there exists $a \in A$ such that $s'/c < a$ meaning $s' > ca$ thus $s'$ cannot be an upper bound on $cA$, and so $s = c \sup A$ is the least upper bound. \item $\sup(cA) = c \inf(A)$ for $c < 0$ \end{enumerate}

Exercise 1.3.5

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Exercise 1.3.5

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