Is my proof for this fact correct?

Question

The thing ought to be proven

Let $a$ and $b$ be nonzero integers that are relatively prime, and let $c$ be an integer. Show that $ax+by=c$ has an integer solution.

My postulated proof that ought to be right

Let $a$ and $b$ be nonzero integers such that $\gcd(a,b)=1$, and let $c$ be an integer. Since $\gcd(a,b)=1$, it follows that the smallest positive linear combination of $a$ and $b$ (which will call $d$) is equal to $\gcd(a,b)=1$, i.e. $d=as+bt=1$. Therefore $a(cs)+b(ct)=c$, hence $ax+by=c$ has an integer solution.

Is my proof correct? How shall I ameliorate it?

Answer 1

Your proof is correct, provided that you are allowed to assume the smallest linear combination of two coprime integers is $1$. If you can, then you're fine.