By Borel's theorem, lattices in simple Lie groups are Zariski dense. I expect that a small (in metric sense) deformation of a lattice in a Lie group is also Zariski dense.

Suppose we have a Zariski dense subgroup $\Gamma$ in a non-compact simple algebraic Lie group $G$. Suppose that we took a subset $\Lambda\subset G$ such that its $r$-neighbourhood contains $\Gamma$. Is the subgroup generated by $\Lambda$ Zariski dense in $G$?

More formally,

CONJECTURE: Let $\Gamma \subset G$ be a Zariski dense subgroup of a non-compact simple algebraic Lie group, and $B_r\subset G$ an open ball in a left-invariant metric. Let $\Lambda\subset G$ be a subset such that $B_r\!\cdot\!\Lambda$ contains $\Gamma$. I conjecture that the group generated by $\Lambda$ is Zariski dense in $G$.

Is it known? I have several ideas how it can be proven, but the "proof" that I have looks ugly, and I have a feeling that the result is known.