The real numbers

A set $S$ of real numbers is bounded above if some number $M$ satisfies $s\le M$ for every $s\in S$; the least such $M$ is the supremum, written $\sup S$. Bounded below and infimum are defined similarly. The completeness axiom states that every nonempty set bounded above has a supremum in $\mathbb{R}$. A number $x$ equals $\sup S$ exactly when $x$ is an upper bound of $S$ and every number smaller than $x$ fails to be an upper bound. To prove $\sup S=x$, show $s\le x$ for all $s\in S$, and for any $\epsilon >0$ find some $s\in S$ with $s>x-\epsilon$. The same test with reversed inequalities works for $\inf S$.