The existence of integers exceeding real numbers motivates the following definitions: * The greatest integer \(n\) less than or equal to \(x\) denoted by \(\lfloor x \rfloor\). * The least integer \(n\) greater than or equal to \(x\) denoted by \(\lceil x \rceil\).
The floor $\lfloor \cdot \rfloor:\mathbb R\to\mathbb Z$ and the ceiling function $\lceil \cdot \rceil:\mathbb R\to\mathbb Z$ are functions from the set $\mathbb R$ of real numbers to the set $\mathbb Z$ of integers, defined by:
\[\begin{array}{rcl} \lfloor x \rfloor=n&\Longleftrightarrow& n\le x < n+1,\\ \lfloor x \rfloor=n&\Longleftrightarrow& x-1 < n \le x,\\ \lceil x \rceil=n&\Longleftrightarrow& x\le n < x+1,\\ \lceil x \rceil=n&\Longleftrightarrow& n-1 < x \le n.\\ \end{array}\]
Definitions: 1
Lemmas: 2 3 4
Proofs: 5 6 7 8 9 10 11
Propositions: 12 13 14 15
Theorems: 16
