Exercise 1.3.2

Question

Give an example of each of the following, or state that the request is impossible.
  \begin{enumerate}[label=(\alph*)] 
  \item A set $B$ with inf $B \geq \sup B$.
  \item A finite set that contains its infimum but not its supremum.
  \item A bounded subset of $\mathbf{Q}$ that contains its supremum but not its infimum.
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item Let $B = \{0\}$ we have $\inf B = 0$ and $\sup B = 0$ thus $\inf B \ge \sup B$.
  \item Impossible, finite sets must contain their infimum and supremum.
  \item Let $B = \{r \in \mathbf{Q} \mid 1 < r \le 2\}$ we have $\inf B = 1 
otin B$ and $\sup B = 2 \in B$.
   \end{enumerate}

Exercise 1.3.2

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Exercise 1.3.2

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