Compact subgroups are contained in open compact subgroups in locally profinite groups

Question

Let $G$ be a totally disconnected, Hausdorff, locally compact group. In the wikipedia page about these groups there is a claim that any compact subgroup of $G$ is contained in some compact open subgroup. Is this true? If so, why?

Answer 1

It seems the following.

By [Pon, Theorem 16], the group $G$ has a base at the unit consisting of its open compact subgroups. In particular, the group $G$ contains a compact open subgroup $K$. Now let $C$ be a compact subgroup of the group $G$. Put $N=\bigcap \{x^{-1}Kx:x\in C\}$. It is easy to check that $N$ is a closed subgroup of the group $K$ (and hence a compact) and $CN$ is a group. The set $CN$ is compact as a product of two compact sets. We claim that $N$ is an open subset of the group $G$. Indeed, let $x\in C$ be an arbitrary element. Since $x^{-1}ex\in K$, and $K$ is open, there exist open neighborhoods $V_x\ni e$ and $W_x\ni x$ such that $W_x^{-1}V_xW_x\subset K$. The family $\{W_x:x\in C\}$ is an open cover of a compact space $C$. Therefore there exists a finite subset $F$ of the group $C$ such that $C\subset\bigcup\{W_x:x\in F\}$. Put $V=\bigcap\{V_x:x\in F\}$. Then $x^{-1}Vx\subset K$ for each element $x\in C$. Thus $V\subset N$. Hence for each element $y\in N$ we have $Vy\subset N$. Therefore, the set $N$ is open. Similarly we can show that the set $CN$ is open. Therefore $CN$ is an open compact subgroup of the group $G$, which contains the group $C$.

References

[Pon] Lev S. Pontrjagin, Continuous groups, 2nd ed., M., (1954) (in Russian).