Compact and totally disconnected implies every open neighborhood of identity contains an open normal subgroup

Statement
Suppose $$G$$ is a profinite group (i.e., it is a compact Hausdorff totally disconnected group) and $$U$$ is an open subset of $$G$$ containing the identity element of $$G$$. Then, $$U$$ contains an open normal subgroup of $$G$$.

Facts used

 * 1) uses::Compact neighborhood of identity in totally disconnected group contains compact open subgroup
 * 2) uses::Intersection of conjugates by compact subset of open neighborhood of identity contains open neighborhood of identity