Hello;

Show that if is a first countable topological group, then there is a sequence of symmetric neighborhoods of the neutral element of such that

(1){ } is a local base at in G

(2) , for every

I understand number 1 is due to first countability of , 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.