Ceiling of Number plus Integer
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Theorem
- $\forall n \in \Z: \ceiling x + n = \ceiling {x + n}$
where $\ceiling x$ denotes the ceiling of $x$.
Proof
| \(\ds \ceiling {x + n} - 1\) | \(<\) | \(\, \ds x + n \, \) | \(\, \ds \le \, \) | \(\ds \ceiling {x + n}\) | Number is between Ceiling and One Less | |||||||||
| \(\ds \leadsto \ \ \) | \(\ds \ceiling {x + n} - n - 1\) | \(<\) | \(\, \ds x \, \) | \(\, \ds \le \, \) | \(\ds \ceiling {x + n} - n\) | |||||||||
| \(\ds \leadsto \ \ \) | \(\ds \ceiling x\) | \(=\) | \(\ds \ceiling {x + n} - n\) | Number is between Ceiling and One Less | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \ceiling {x + n}\) | \(=\) | \(\ds \ceiling x + n\) | adding $n$ to both sides |
$\blacksquare$