Real Number is Ceiling minus Difference

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

Then:
 * $$x = n - t: n \in \Z, t \in \left[{0 \, . \, . \, 1}\right) \iff n = \left \lceil {x}\right \rceil$$

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

Proof

 * Let $$x = n - t$$, where $$t \in \left[{0 \, . \, . \, 1}\right)$$.

Now $$1 - t > 0$$, so $$n - 1 < x$$.

Thus $$n = \inf \left({\left\{{m \in \Z: m \ge x}\right\}}\right) = \left \lceil {x}\right \rceil$$.


 * Now let $$n = \left \lceil {x}\right \rceil$$.

From Ceiling Minus Real Number, $$\left \lceil {x}\right \rceil - x \in \left[{0 \,. \, . \, 1}\right)$$.

Here we have $$\left \lceil {x}\right \rceil = n$$.

Thus $$\left \lceil {x}\right \rceil - x \in \left[{0 \,. \, . \, 1}\right) \implies n - x = t$$, where $$t \in \left[{0 \, . \, . \, 1}\right)$$.

So $$x = n - t$$, where $$t \in \left[{0 \,. \, . \, 1}\right)$$.