In theorem 6.36. in the book compact Lie groups by Sepanski, the argument used to prove that the Weyl group is finite is the following
Since the set of roots $\Delta(\mathfrak{g}_\mathbb{C}, \mathfrak{t}_\mathbb{C})$ is finite and since the Weyl group $W$ acts faithfully on $\Delta(\mathfrak{g}_\mathbb{C}, \mathfrak{t}_\mathbb{C})$, then $W$ is finite.
Why this is true ?
Since $W$ acts faithfully on a finite set, $W$ is a subgroup of the symmetric group on the set of roots. As a subgroup of a finite group, $W$ is finite.