Ceiling of Non-Integer

Theorem
Let $x \in \R$ be a real number.

Let $x \notin \Z$.

Then:
 * $\left \lceil{x}\right \rceil < x$

where $\left \lceil{x}\right \rceil$ denotes the ceiling of $x$.

Proof
From Ceiling is between Number and One More:
 * $\left \lceil{x}\right \rceil \ge x$

From Integer equals Ceiling:
 * $x = \left \lceil {x} \right \rceil \iff x \in \Z$

But we have $x \notin \Z$.

So:
 * $\left \lceil {x} \right \rceil \ne x$

and so:
 * $\left \lceil {x} \right \rceil > x$

Also see

 * Floor of Non-Integer