How would I prove the following?
Show that the assertion of the Bolzano-Weierstrass Theorem is equivalent to the Completeness Axiom for real numbers.
I know that the Nested Set Theorem implies the Bolzano-Weierstrass Theorem, and that the Completeness Axiom implies the Nested Set Theorem, so I understand how Completeness Axiom $\Rightarrow$ Bolzano-Weierstrass. How do I show that the Bolzano-Weiertsrass Theorem implies the Completeness Axiom?
You can try to prove Nested Interval Principle using Bolzano Weierstrass Theorem. Just choose one point from each interval in the sequence of nested intervals such that these points are distinct and then the chosen infinite set of such points possesses an accumulation point $c$. This $c$ must lie in all the nested intervals (why?)
Next show that Nested Interval Principle implies Completeness Axiom. Let $A$ be a set bounded above by $b$ and let $a$ be some number less than some element of $A$. Consider the interval $[a, b] $. If the midpoint $c=(a+b) /2$ is an upper bound for $A$ then choose $a_{1}=a,b_{1}=c$ otherwise choose $a_{1}=c,b_{1}=b$. Using same procedure get $a_{2},b_{2}$ from $a_{1},b_{1}$ and repeat the process indefinitely to get a sequence of nested intervals. There is a unique $c$ which lies in all these intervals and one can easily show that $c=\sup A$ (how?)
Not an Answer "In their attempt at providing rigorous proofs of some basic facts about continuity, Bernard Bolzano (1781–1848) and Augustin Louis Cauchy (1789–1857) made use of what we now call the Cauchy Completeness Theorem, though they could not prove it because they lacked the axiomatic properties of the real numbers. Bolzano did provide a proof that the Cauchy Completeness Theorem implied the Least Upper Bound Property, using the idea of bisection. Cauchy’s proof of the Intermediate Value Theorem relied implicitly upon the Monotone Con- vergence Theorem, and explicitly on the fact that a continuous function works nicely with respect to convergent sequences. In the 1860s Karl Weierstrass (1815–1897) used a bisection argument similar to Bolzano’s to prove a version of what we now call the Bolzano–Weierstrass Theorem for bounded infinite sets.
Richard Dedekind (1831–1916), using his construction of the real numbers from the rational numbers in Stetigkeit und irrationale Zahlen of 1872 (originally formulated in lectures in 1858), provided what was probably the first rigorous proof of the Monotone Convergence Theorem. Such a proof was not possible without a rigorous treatment of the real numbers."
The Real Numbers and Real Analysis by Ethan D. Bloch
