Image of projective representations of Weil group

Question

Let $F$ be a number field and let $\sigma : G_{F} = \mathrm{Gal}(\overline{F}/F) \to \mathrm{GL}_{2}(\mathbb{C})$ be a continuous representation of Galois group. Since $G_F$ is compact, the image of $\sigma$ in $\mathrm{PGL}_{2}(\mathbb{C})$ is finite, and the classification for the subgroups is known by Klein. In Gelbart's article on Langlands-Tunnell theorem, it saids that the same thing holds for the representations of Weil group $W_{F}$ (it has same classification on projective image). However, since $W_F$ is not compact, I can't show that the image is finite. Could anyone help?