The left <cosets> of an <open subgroup> $H$ form an <open cover> of the <compact group>. A finite subfamily therefore covers the <group>. Distinct <cosets> are disjoint, so every <coset> must occur in this finite subfamily. Hence \b[$[G:H]<\infty$].

Solution

Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 5 / 1 / ii / Solution

 Solution (source code)