Exercise 3.4.3

Question

Review the portion of the proof given in Example 3.4.2 and follow these steps to complete the argument.
  \begin{enumerate}[label=(\alph*)] 
  \item Because $x \in C_{1}$, argue that there exists an $x_{1} \in C \cap C_{1}$ with $x_{1} 
eq x$ satisfying $\left|x-x_{1}ight| \leq 1 / 3$.
  \item Finish the proof by showing that for each $n \in \mathbf{N}$, there exists $x_{n} \in C \cap C_{n}$, different from $x$, satisfying $\left|x-x_{n}ight| \leq 1 / 3^{n}$.
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item Noting that $C_1$ is the union of disjoint intervals of length $1/3$, and that $C_2$ divides each interval in $C_1$ into two, consider the intervals $[a, b] \subseteq C_1$ and $[c, d] \subseteq C_2$ that $x$ is in. Then choose $x_1$ to be any other point $c \in C \cap ([a, b] \backslash [c, d])$ - i.e. it shares an interval with $x$ in $C_1$ but is in a different interval in $C_2$; therefore it must be within $1/3$ of $x$ but is different from $x$.
  \item Identical argument to part (a), replacing $C_1$ with $C_n$, $C_2$ with $C_{n+1}$, $1/3$ with $1/3^n$, and $x_1$ with $x_n$.
   \end{enumerate}

Exercise 3.4.3

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Exercise 3.4.3

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