Proof about GCD's Prove that if $a, b$ and $c$ are integers with $b \neq 0$ and $a=bx+cy$ for some integers $x$ and $y$, then $\text{gcd}(b,c) \le  \text{gcd}(a,b).$

I don't understand how to show (b,c) is less than (a,b)?

$\ \begin{eqnarray}\color{}{(b,c)}&&\mid\, \color{sienna}b,\ \ \ \color{sienna}c \\ \Rightarrow\ (b,c)&&\mid \,\color{sienna}b\,x\!\!+\!\!\color{sienna}c\,y=\color{#c00}a\!\end{eqnarray}\bigg\rbrace\  $ so $\ \bigg\lbrace\begin{eqnarray}(b,c)&\mid&\ \color{#c00}a,\,b\\ \,\Rightarrow (b,c) &\color{#0a0}\le& (a,b) \end{eqnarray}\bigg\rbrace\ $ by $\ \bigg\lbrace\, \begin{eqnarray} &&c\rm ommon\ divisor \\ \color{#0a0}\le &&{\it greatest}\rm\,\  common\ divisor\end{eqnarray}$
