I have currently read a proof (existence of sections for pro-finite groups (in the book profinite groups of Ribes)) and I did not understand the following two facts used (without mentioning any details):
1) We are given a pro-finite group $G$ (that is, a compact, Hausdorff and totally disconnected topological group)and two closed subroups $K\leq H$. We first assume that the quotient $H/K$ is finite. Then there exists an open subgroup $U$ of $G$ such that $U\cap H\subset K$. I do not understand how to find such an open subgroup $U$.
2) Next we construct a closed subgroup $T$ of $G$ such that $K\leq T\leq H$ and we assume that $T\neq K$. Then there exists an open subgroup $U$ of $G$ such that with $K\subseteq (U\cap T)$. Again I do not understand how to find such an open subgroup $U$.
Any help is much appreciated. Thanks!