Let $G$ be a Lie group with left translation action:
$L: G \times G \rightarrow G$ $\quad$ $(g,h) \rightarrow L_g(h) := gh$
I want to show that this action is transitive,free, and proper.
Transitive and free follow from definition more or less.
However, I am having trouble showing it is proper.
This is the definition of a proper action that I am given:
The action $\phi:G \times M \rightarrow M$ of a group $G$ on a manifold $M$ is proper if, whenever the sequences $\{x_n\}$ and $\{g_nx_n\}$ converge in $M$, the sequence $\{g_n\}$ has a convergent subsequence in $G$.
It is clear to me that if $G$ is a compact Lie group, then the action is proper. This is so because in a compact group, every sequence $\{g_n\}$ has a convergent subsequence. But we are not given that the group $G$ is compact. My question is, how do we show this for a generic Lie group $G$?
Any hints/suggestions are most welcomed.
Best Answer
Hint : if $x_n\to x, g_nx_n \to y$, then $g_n = g_nx_nx_n^{-1}\to ?$