Ceiling is between Number and One More

Theorem

$x \le \ceiling x < x + 1$

where $\ceiling x$ denotes the ceiling of $x$.


Proof

From Number is between Ceiling and One Less:

$\ceiling x - 1 < x \le \ceiling x$

Thus by adding $1$:

$x + 1 > \paren {\ceiling x - 1} + 1 = \ceiling x$

So:

$x \le \ceiling x$

and:

$\ceiling x < x + 1$

as required.

$\blacksquare$


Sources

  • 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.4$: Integer Functions and Elementary Number Theory
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.