Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 5 / 1 / ii / Solution

The cosets of an open subgroup $H$ form an open cover of the compact space $G$. A finite subcover exists. Since distinct cosets are disjoint, that subcover must contain every coset. Thus $[G:H]$ is finite. Neither normality nor Hausdorffness is needed for this argument.