I am currently doing self-study on profinite groups and I'm stuck trying to prove the following lemma.
If a topological group $G$ is compact and totally disconnected, then the open normal subgroups of $G$ intersect in the trivial subgroup $\{\,1_{G}\,\}$.
While I hope that mere hints will allow me to see how to prove this, I think I may need someone to just spell-it-out for me. I've been racking my brain as to how I could prove this and I've made little progress. Either way, any help would be much appreciated.