# Smaller neighborhood around the identity of Lie Group

This is the problem 7-6 of Lee's Introduction to Smooth Manifolds (2nd edition):

Suppose G is a Lie group and U is any neighborhood of the identity. Show that there exists a neighborhood V of the identity such that $$V \subset U$$ and $$gh^{-1} \in U$$ whenever $$g, h \in V$$.

How do I approach this problem? I tried using the smoothness of $$(g,h) \mapsto gh^{-1}$$ or the open subgroup generated by $$U$$, but it didn't get me anywhere.

You don’t need anything fancy here. Just use continuity and pick $$V$$ carefully.
The map $$f : G \times G \to G$$ defined as $$f(g,h) = gh^{-1}$$ is a smooth map. Suppose $$U$$ is a neighbourhood of the identity $$e\in G$$. By continuity, $$W = f^{-1}(U)$$ is open in $$G \times G$$. Since $$(e,e) \in W$$, there are neighbourhoods $$W_1,W_2 \subseteq G$$ containing $$e\in G$$ such that $$W_1\times W_2 \subseteq W$$. Choose $$V$$ as
$$V = (W_1 \cap W_2) \cap U$$
So $$V \times V \subseteq W=f^{-1}(U)$$ implies $$f(V \times V) \subseteq U$$. I.e., $$\forall g,h \in V$$, we have $$f(g,h)=gh^{-1} \in U$$.