Hello;

Show that if $\displaystyle G$ is a first countable topological group, then there is a sequence of symmetric neighborhoods $\displaystyle (U_{n})_{n}$ of the neutral element $\displaystyle e$ of $\displaystyle G$ such that

(1){ $\displaystyle {U_{n}:n\in Z }$} is a local base at $\displaystyle e$ in G

(2) $\displaystyle U_{n+1}^{3}\subset U_{n}$, for every $\displaystyle n$

I understand number 1 is due to first countability of $\displaystyle G$, but how can we get (2). Please guide me. Every comment or guidance is highly appreciated

Recall that a neighborhood of the identity element is said to be symmetric if it equal to its inverse.