Sorry for the (edited) duplicate from overflow but I did not receive any answer there.
For an action by a Lie group $G$ on a second countable and Hausdorff differentiable manifold $M$ to be proper, it suffices to show that the (extended) map $G \times M \rightarrow M \times M$ is closed with respect to prescribed topologies on these spaces and that the fibers are compact. No local compactness is assumed here for either $G$ or $M$ though. As mentioned in many texts, one way to proceed is to verify that given a sequence $\{g_n\}$ in $G$ and a converging sequence $\{x_n\}$ in $M$, if the sequence $\{g_n \cdot x_n\}$ converges in $M$, then $\{g_n\}$ contains a converging subsequence.
That this formulation is equivalent, under appropriate assumptions, to some other forms of definition for proper group action is clear to me. However, I have searched places I could think of including posts at this site but could not seem to find any specific example that actually uses this sequence characterization to prove properness of a group action.
I am just beginning to learn this material, and understand that different techniques are called for in different scenarios to extract a converging subsequence. But could someone kindly provide or point me to some less trivial and concrete example(s) illustrating necessary procedures to arrive at a converging subsequence in the differentiable context?
Thanks a lot!