# Existence of an open normal subgroup of a neighborhood of 1 in a compact Hausdorff and totally disconnected topological group

Let $G$ a compact Hausdorff and totally disconnected topological group. Then every neighborhood of 1 contains an open normal subgroup of finite index in $G$. I need this lemma to prove that every compact Hausdorff and totally disconnected topological group is a profinite group. I am trying but I can not prove it. Any suggestions?