Exercise 2.4.4

Question

\begin{enumerate}[label=(\alph*)] 
  \item In Section 1.4 we used the Axiom of Completeness (AoC) to prove the Archimedean Property of $\mathbf{R}$ (Theorem 1.4.2). Show that the Monotone Convergence Theorem can also be used to prove the Archimedean Property without making any use of AoC.
  \item Use the Monotone Convergence Theorem to supply a proof for the Nested Interval Property (Theorem 1.4.1) that doesn't make use of AoC.

These two results suggest that we could have used the Monotone Convergence Theorem in place of $\mathrm{AoC}$ as our starting axiom for building a proper theory of the real numbers.
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item MCT tells us $(1/n)$ converges, obviously it must converge to zero therefore we have $|1/n - 0| = 1/n < \epsilon$ for any $\epsilon$, which is the Archimedean Property.
  \item We have $I_n = [a_n, b_n]$ with $a_n \le b_n$ since $I_n
e \emptyset$. Since $I_{n+1} \subseteq I_n$ we must have $b_{n+1} \le b_n$ and $a_{n+1} \ge a_n$ the MCT tells us that $(a_n) 	o a$ and $(b_n) 	o b$. by the Order Limit Theorem we have $a \le b$ since $a_n \le b_n$, therefore $a \in I_n$ for all $n$ meaning $a \in \bigcap_{n=1}^\infty I_n$ and thus $\bigcap_{n=1}^\infty I_n 
e \emptyset$.
   \end{enumerate}

Exercise 2.4.4

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Exercise 2.4.4

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