Integer Multiple of Integer Combination is Integer Combination

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 $b$.

Let $u \in S$.

Let $n \in \Z$.

Then $n u \in S$.

Proof
Let $u = ax + by$ where both $x$ and $y$ are integers.

Then:

As Integer Multiplication is Closed, both $n x$ and $n y$ are integers.

Hence the result.