A simple proof of a theorem of Karrass and Solitar
  In this note we give a particularly short and simple proof of the following theorem of Karrass and Solitar. Let $H$ be a finitely generated subgroup of a free group $F$ with infinite index $[F:H]$. Then there is a nontrivial normal subgroup $N$ of $F$ such that $N\cap H = \{1\}$. 
Delaram Kahrobaei (University of St Andrews)