Existence of q for which j - qk is Positive

Theorem
Let $j, k \in \Z$ be integers such that $k > 0$.

Then there exist $q \in \Z$ such that $j - q k > 0$.

Proof
Let $q = -\size j - 1$.

Then:

We have that:
 * $\forall j \le 0: j + \size j = 0$

and:
 * $\forall j > 0: j + \size j = 2 j$

So:
 * $j - q k \ge k$

and as $k > 0$ the result follows.