The Well Ordering Principle

Theorem 0.1 [Well Ordering Principle]

Let $E$ be a non-empty subset of $\mathbb{N}$. Then, $\inf(E)$ exists and $\inf(E) \in E$.

Proof

Consider the set $-E$. $-E$ is bounded above by $0$. Since is it non-empty and bounded, then by the completeness axiom, the supremum of $-E$ exists. Moreover, by Lemma/Theorem 1.15, we know that $\sup(-E) \in -E$. Hence

$$ \inf(E) = -\sup(-E) $$

exits and is in $E$. $\ \blacksquare$

References

Lecture Notes by Professor Chun Kit Lai