Floor of m+n-1 over n

Theorem
Let $m, n \in \Z$ such that $n > 0$.

Then:
 * $\left \lfloor{\dfrac {m + n - 1} n}\right \rfloor = \left \lceil{\dfrac m n}\right \rceil$

The identity does not necessarily apply for $n < 0$.

Proof
First let $n > 0$ as stated.

Let $x = \left \lfloor{\dfrac {m + n - 1} n}\right \rfloor$.

Then:

Setting $m = 1, n = -2$ we have:

Thus, as stated, it is confirmed that the identity does not hold for $n < 0$.

It is noted that when $n = 0$ the expressions on either side are not defined.