Continuum Property implies Well-Ordering Principle

Theorem
The Continuum Property of the Positive Real Numbers implies the Well-Ordering Principle of the Natural Numbers.

Proof
Suppose that $A \in \R^+$.

Then the well-ordering principle is true for all subsets $S \subseteq \N$.

For if $S$ has a least element, then the well-ordering principle is true for $S$.

If $T \subseteq \N$, then $T$ has a least element.

For if not, then suppose that $T \subseteq \N$ has no least element.

Then for every $t \in T$, there exists a $u \in T$ such that $u < t$.

The entire class $\R^+$ of positive real numbers has the Continuum Property.

For every $t \in T$, let $a_t= \dfrac {A}{2^t}$.

$a_t$ can never be greater than $A$, therefore $a_t$ has an upper bound, and therefore a least upper bound $B$.

Because $\dfrac {7B}{8}$ is not an upper bound on $a_t$, there exists $m \in T$ such that $a_m > \dfrac {7B}{8}$.

Then there is $n \in T$ such that $n < m$.

Let $n = m - p$.

Then

Therefore $a_n > B$, even though $B$ is the least upper bound: which is impossible.

Hence every $T \subseteq \N$ has a least element, and therefore the Well-Ordering Principle is true for all subsets $S \subseteq \N$.

Also see

 * Equivalence of Well-Ordering Principle and Induction