proof that if a|b and b|c then a|c

Question

Just wanted some feed back on the following proof

"if $a$ divides $b$ and $b$ divides $c$ then $a$ divides $c$"

I came up with this:

If $a|b$ then there exist some $x$ that $a * x = b$ and if $b|c$ there exist some integer $y$ that $b * y =c$ therefore if $a|c=z$ and $z=xy$ then $(a|b)(b|c)=a|c )$.

Please let me know if I got this correct and I also wanted to know if there are more than one way to prove this statement, Thank you.

Answer 1

$b=ka$ and $c=k'b$ implies $c=kk'a$ and thus $a\mid c$.