Exercise 3.2.4

Question

Let $A$ be nonempty and bounded above so that $s=\sup A$ exists.
  \begin{enumerate}[label=(\alph*)] 
  \item Show that $s \in \overline{A}$.
  \item Can an open set contain its supremum?
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item Since every $s - \epsilon$ has an $a \in A$ with $a > s-\epsilon$ we can find $a \in V_\epsilon(s)$ for any $\epsilon > 0$, meaning $s$ is a limit point of $A$ and thus contained in $\overline{enumerate}

Exercise 3.2.4

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Exercise 3.2.4

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