Even before we know what the divides symbol means, we can set up a direct proof for this statement. It will go something like this: Let \(a\text{,}\) \(b\text{,}\) and \(c\) be arbitrary integers. Assume that \(a|b\) and \(b|c\text{.}\) Dot dot dot. Therefore \(a|c\text{.}\)
How do we connect the dots? We say what our hypothesis (\(a|b\) and \(b|c\)) really means and why this gives us what the conclusion (\(a|c\)) really means. Another way to say that \(a|b\) is to say that \(b = ka\) for some integer \(k\) (that is, that \(b\) is a multiple of \(a\)). What are we going for? That \(c = la\text{,}\) for some integer \(l\) (because we want \(c\) to be a multiple of \(a\)). Here is the complete proof.
Let \(a\text{,}\) \(b\text{,}\) and \(c\) be integers. Assume that \(a|b\) and \(b|c\text{.}\) In other words, \(b\) is a multiple of \(a\) and \(c\) is a multiple of \(b\text{.}\) So there are integers \(k\) and \(j\) such that \(b = ka\) and \(c = jb\text{.}\) Combining these (through substitution) we get that \(c = jka\text{.}\) But \(jk\) is an integer, so this says that \(c\) is a multiple of \(a\text{.}\) Therefore \(a|c\text{.}\)