Good morning, my answer is related to the one introduced here
Convergence of images of a sequence of converging continuous functions
but with stronger hypotheses. Suppose you have a sequence of invertible functions $f_n$ that converges uniformly to $f$ on a limited domain $X$. Moreover $f_n$ and $f$ are Lipschitz and the L. constants $L_n$ of $f_n$ converges to the one of $f$.
If $A$ is an open set, can we prove that
$$ |f_n(A) \setminus f(A)| \to 0 ? $$
Do you know any references?