Generic elements of a Zariski-dense subgroup form an open subset

Let $G$ be a semi-simple algebraic group over a finitely generated field $K$ of characteristic zero, and let $\Gamma \subset G(K)$ be a finitely generated Zariski-dense subgroup. In this note we prove that the set of $K$-generic elements of $\Gamma$ (whose existence was established earlier in [PR301]) is open in the profinite topology of $\Gamma$. We then extend this result to the fields of positive characteristic, and also prove the existence of generic elements in this case.