Floor of Number plus Integer
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Theorem
- $\forall n \in \Z: \floor x + n = \floor {x + n}$
where $\floor x$ denotes the floor of $x$.
Proof
| \(\ds \floor {x + n}\) | \(\le\) | \(\, \ds x + n \, \) | \(\, \ds < \, \) | \(\ds \floor {x + n} + 1\) | Definition of Floor Function | |||||||||
| \(\ds \leadsto \ \ \) | \(\ds \floor {x + n} - n\) | \(\le\) | \(\, \ds x \, \) | \(\, \ds < \, \) | \(\ds \floor {x + n} - n + 1\) | |||||||||
| \(\ds \leadsto \ \ \) | \(\ds \floor x\) | \(=\) | \(\ds \floor {x + n} - n\) | Definition of Floor Function | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \floor {x + n}\) | \(=\) | \(\ds \floor x + n\) | adding $n$ to both sides |
$\blacksquare$