Equivalence of Definitions of Ceiling Function

Theorem
Let $x$ be a real number.

Definition 1 equals Definition 2
Follows from Infimum of Set of Integers equals Smallest Element.

Definition 1 equals Definition 3
Let $S$ be the set:
 * $S = \left\{ {m \in \Z: m \ge x}\right\}$

Let $n = \inf S$.

By Infimum of Set of Integers is Integer, $n \in \Z$.

By Infimum of Set of Integers equals Smallest Element, $n\in S$.

Because $n\in S$, we have $n \geq x$.

Because $n-1 < n$, we have by definition of supremum:
 * $n-1 \notin S$

Thus $n-1 < x$.

Thus $n$ is an integer such that:
 * $n-1 < x \leq n$

So $n$ is the ceiling function by definition 3.

Definition 3 equals Definition 2
Let $n$ be an integer such that:
 * $n-1 < x \leq n$

We show that $n$ is the smallest element of the set:
 * $S = \left\{ {m \in \Z: m \ge x}\right\}$

Let $m \in \Z$ such that $m \ge x$.

We show that $n\leq m$.

$m < n$.

By Weak Inequality of Integers iff Strict Inequality with Integer minus One:
 * $m \le n - 1$

and so from the definition of $g$ it follows that $m < x$.

By Proof by Contradiction it follows that $m \ge n$.

Because $m \in S$ was arbitrary, $n$ is the smallest element of $S$.

Thus $n$ is the ceiling function by definition 2.

Also see

 * Equivalence of Definitions of Floor Function