Show that the intersection of 2 subgroups of finite index also has finite index.

Proof.

Let $\displaystyle \{S_{a} \} _{a \in I} $ and $\displaystyle \{ T_{b} \} _{b \in J} $

The intersection of them, well, since the index of both are finite, isn't the intersection $\displaystyle \bigcap _{c \in I \cup J } \{U _{c} \} $ well, then, the index has to be finite.

But I'm not sure if I write this up correctly.

Thanks.