Ceiling is between Number and One More
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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