Set of Integer Combinations includes Zero

Lemma

Let $a, b \in \Z$ be integers.

Let $S = \left\{{ax + by: x, y \in \Z}\right\}$ be the set of integer combinations of $a$ and $b$.

Then $0 \in S$.

Proof

By setting $x = 0$ and $y = 0$:

$a \cdot 0 + b \cdot 0 = 0$

$\blacksquare$