If $G$ be a locally compact topological group. Show that $G$ is paracompact.
Note: If we restrict $G$ to be locally compact, connected topological group, this problem becomes easier by constructing a sequence of sets $U_{n+1}=\bar{U}_{n}.U_1$ where $U_1$ is a neighborhood of $e$ having compact closure. However, I feel that in this proof I still have not used the full strength of local compactness (only need it to know the existence of $U_1$). In Munkres' book, he stated that if we remove the condition of connectedness, the theorem still remains true. But I have not figured it out yet. Does anyone have any idea?
You are almost there, Let $H= \cup_n \overline U_n$, then $H$ is open and paracompact since it is a countable union of compact sets. $G$ is the union of all the cosets of $H$, this implies that $G$ is paracompact.