Given a finite subset $S$ not containing the identity element in a residually finite group $G$, does there always exist a normal subgroup of $G$ which has finite index (in $G$) and 
which avoids $S$? (If $S$ is a singleton, this is of course the definition of a residually finite group.)



Finite subsets in residually finite groups
Yes, take the intersection of the normal subgroups $N\_1, N\_2, ..., N\_k$ of finite index avoiding elements $x\_1,x\_2,...,x\_k$ of your set. It is normal and of finite index (at most the product of indices of $N\_i$). 


