Exercise 3.3.12

Question

Using the concept of open covers (and explicitly avoiding the Bolzano-Weierstrass Theorem), prove that every bounded infinite set has a limit point.

Ulisse Mini · Accepted Answer

Let $A$ be an infinite set bounded by $M$ (i.e. $|x|<M$ for all $x \in A$),
  suppose for contradiction that $A$ has no limit points,
  meaning there exists an $\epsilon > 0$ such that $V_\epsilon(x) \cap A = \{x\}$ for all $x \in A$.

This immediately implies there are only a finite number of sets in our cover, otherwise the union would be unbounded. Contradiction.

extbf{TODO } Finish tikz picture and proof (its visually obivous)

\begin{figure}[!h]
    \centering
    \begin{tikzpicture}
      \draw[thick] (-6,0) -- (6, 0);
      \draw[thick] (-5, 0.2) .. controls (-5.1, 0) .. (-5, -0.2);
      % \draw[thick, ->] (-5,0.2) arc (0:220:1);
    \end{tikzpicture}
  \end{figure}

Exercise 3.3.12

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Exercise 3.3.12

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