Let $\displaystyle a, b$ be integers, not both equal to zero, $\displaystyle d = min\{ax + by|x, y \in Z\}$. Prove $\displaystyle d = gcd(a, b)$

Attempt: I can do final part of proof: If $\displaystyle d | a$ and $\displaystyle d | b$, then $\displaystyle d | gcd(a, b)$. From $\displaystyle \exists x_0, y_0$, s.t. $\displaystyle d = ax_0 + by_0$ we have $\displaystyle gcd(a,b) | d$. From $\displaystyle gcd(a,b) | d$ and $\displaystyle d | gcd(a,b)$ finally $\displaystyle d = gcd(a,b)$.

But I can't prove that $\displaystyle d | a$ and $\displaystyle d | b$.

Can anyone help please? Thank you!