How to prove that if $c$ divides $a$, and $c$ divides $b$, then $c \leq d$?

Question

Let $a,$ $b,$ and $d$ be positive integers. Suppose that $d$ divides $a$ and $d$ divides $b,$ and also for every integer $c,$ if $c$ divides $a$ and $c$ divides $b,$ then $c$ divides. Then we say that $d$ is de GCD of $a$ and $b.$ If $c$ divides $a$ and $c$ divides $b,$ prove that $c\leq d.$

Answer 1

Given that $c$ and $d$ divide $a$ and $b$, they are common divisors. Since $d$ is the Greatest Common Divisor, it is automatically greater than $c$, unless they are equal.

So, $d\ge{c}$