08 - Completeness axiom for the real numbers

Completeness axiom for the real numbers

Let $S$ be a

1. Non-empty subset of $\mathbb{R}$
2. Bounded above

Then $S$ has a supremum

Note that non-empty is very important otherwise every real number is an upper bound

$\mathbb{Q}$ is not complete (with the ordering inherited in $\mathbb{R}$)