We have arbitrary $a, b, c \in \mathbb N.$
Let $d$ be the biggest natural number, such that $d\mid a,\; d\mid b,\;$ and $\;d\mid c.$
Prove that $d = \gcd ( \gcd (a,b), c).$
I think that considering that d is divisible by a and b, so does their GCD have to be and since c is also a divisor of d, GCD of these two numbers has to be d.
Am I right? How do I formulate this into a real proof? If it's even correct.