# 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

$\left \lceil{x}\right \rceil \ge x$
$x = \left \lceil {x} \right \rceil \iff x \in \Z$

But we have $x \notin \Z$.

So:

$\left \lceil {x} \right \rceil \ne x$

and so:

$\left \lceil {x} \right \rceil > x$

$\blacksquare$