**Definition** The *ceiling function* of a real number $x$, written $\lceil x \rceil$, is defined as $\lceil x \rceil=\text{the smallest integer $n \in \mathbb{Z}$ such that $n \geq x$ }$ **Proof of existence** *Similar to that of [[Floor Function]].* Let $S= \{ m \in \mathbb{Z}: m \geq x \}$. By definition, $S$ is bounded below. Also, by [[Archimedean Property of Real Numbers]], $\exists m \in \mathbb{Z}$ such that $m>x \implies m \in S$. Hence by, [[Greatest Lower Bound Property]], $\exists r:= \inf S$. Since $r$ is the greatest lower bound, $\exists n \in S$ such that $n<r+1$. So $n-1<r \implies n-1 \not \in S \implies n-1<x$. So we have $n-1<x \leq n$, which with $\lceil x \rceil:=n$, rearranges to give $x\leq\lceil x \rceil<x+1$.