so far I understand about the statement: let $p_i,i=1\dots,n$ has non trivial stabilizers i.e $S_{p_1}=\{g:g.p=p, g\in G\}\neq\{e\}$, is non trivial subgroup of $G$ for $p_1$ and so forth upto $p_n$ we will get $S_{p_n}$,so we need to show $\{p_1,\dots,p_n\}$ is discrete.
could any one make me understand the 2nd line of the proof? and in 3rd line $g$ is continous, how come a point $g\in G$ is continuous?