Connected implies no proper closed subgroup of finite index

Statement
In a connected topological group, there cannot be any proper closed subgroup of finite index (i.e., a  closed subgroup that is also a  subgroup of finite index).

Related facts

 * Connected implies no proper open subgroup
 * Open subgroup implies closed
 * Closed subgroup of finite index implies open
 * Compact implies every open subgroup has finite index

Facts used

 * 1) uses::Closed subgroup of finite index implies open

Proof
The proof is direct from Fact (1), and the observation that the existence of a proper subgroup (and hence a proper nonempty subset) that is both closed and open means that the group is not connected.