Exercise 1.3.11

Question

Decide if the following statements about suprema and infima are true or false. Give a short proof for those that are true. For any that are false, supply an example where the claim in question does not appear to hold.
  \begin{enumerate}[label=(\alph*)] 
  \item If $A$ and $B$ are nonempty, bounded, and satisfy $A \subseteq B$, then $\sup A \leq$ $\sup B .$
  \item If $\sup A<\inf B$ for sets $A$ and $B$, then there exists a $c \in \mathbf{R}$ satisfying $a<c<b$ for all $a \in A$ and $b \in B$.
  \item If there exists a $c \in \mathbf{R}$ satisfying $a<c<b$ for all $a \in A$ and $b \in B$, then $\sup A<\inf B$.
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] \item True. We know $a \le \sup A$ and $a \le \sup B$ since $A \subseteq B$. since $\sup A$ is the least upper bound on $A$ we have $\sup A \le \sup B$. \item True. Let $c = (\sup A + \inf B)/2$, $c > \sup A$ implies $a < c$ and $c < \inf B$ implies $c < b$ giving $a < c < b$ as desired. \item False. consider $A = \{x \mid x < 1\}$, $B = \{x \mid x > 1\}$, $a < 1 < b$ but $\sup A ot < \inf B$ since $1 ot < 1$. \end{enumerate}

Exercise 1.3.11

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Exercise 1.3.11

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