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 $$\lceil x \rceil$$.

# Definition: Floor and Ceiling Functions

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

Github: ### References

#### Bibliography

1. Graham L. Ronald, Knuth E. Donald, Patashnik Oren: "Concrete Mathematics", Addison-Wesley, 1994, 2nd Edition