Compactly generated subgroups of locally compact Hausdorff groups

Question

I am reading a book where it is mentioned that any locally compact Hausdorff group is union of subgroups which are compactly generated. For this they take an open symmetric neighbourhood $V$ of identity and consider $G_V:=\cup_{n=1}^\infty V^n$ Clearly, $G_V$ is an open subgroup. How can we show that $G=\cup_{V}G_V$ ?

Answer 1

I guess that by a compactly generated group is meant a group generated by a compact neighborhood of the identity. Any locally compact group $G$ has such a neighborhood $U$ and so each element $x$ of $G$ is contained in a subgroup of $G$ generated by a compact neightborhood $U\cup\{x\}$ of the identity.