Let $G$ be a compact group acting continuously and freely on a compact Hausdorff space $X$. Assume that $(g_\lambda)_{\lambda\in\Lambda}$ and $(x_\lambda)_{\lambda\in\Lambda}$ are nets in $G$ and $X$ respectively s.t. $(x_\lambda)_{\lambda\in\Lambda}$ converges to some $x\in X$ and $(g_\lambda\cdot x_\lambda)_{\lambda\in\Lambda}$ converges (in $X$) to $x$. Does it follow that $(g_\lambda)_{\lambda\in\Lambda}$ converges to $e$, the unit of the group $G$?
Another thing is that from compactness of $G$, there exists a convergent subnet of $(g_\lambda)_{\lambda\in\Lambda}$, and then using freeness of the action, one can check easily that the subnet converges to $e$, but of course- it is not enough.
Thanks