$V = W \cap W^{-1}$ a symmetric neighborhood such that $V\cdot V \subset U$.

Question

This is an answer from another question.

Use that the multiplication is continuous and $1\cdot 1=1$. Let $U$ be a neighborhood of $1$. By continuity there are neighborhoods $A,B \ni 1$ such that $AB \subset U$. Let $W= A \cap B$ and then $V:= W \cap W^{-1}$ is a symmetric neighborhood of $1$ such that $V \cdot V \subset U$.

How is $V\cdot V \subset U$?

Answer 1

You have that $W\subseteq A\cap B$ and $A\cdot B \subseteq U$. Hence, $W\cdot W \subseteq A\cdot B \subseteq U$.

Moreover $V\subseteq W$ so $V\cdot V\subseteq W\cdot W \subseteq U$.