Let $K$ be a compact set in a euclidean complex space (say $\Bbb C^n$) and $f=(f_1,f_2):U\hookrightarrow\Bbb C^2$ is an holomorphic embedding, where $U$ is an open neighborhood of $K$. Then there exists $\eta>0$ such that if $g:U\to\Bbb C^2$ is an holomorphic map with $$ \|f-g\|_K<\eta $$ then $g$ is an embedding on a suitable $\widetilde K\subset K^{\circ}$.
Can somebody give a reference for this result please? I am writing a paper and I can't include such a proof, but I have to attach a reference.
Thank you very much.
EDIT: Since we work on a compact set, it is enough to prove that $g$ is an injective immersion on a certain $\widetilde K\subset K^{\circ}$.
Now $f$ injective immersion means \begin{align} &f\;\;\; \mbox{injective on}\;\;U\\ &(f_1'(p),f_2'(p))\neq(0,0)\;\;\;\forall p\in U. \end{align}
the latter one being equivalent to $df_p$ with maximum rank at each point $p$ (wlog $n\ge2$).
So isn't the conclusion straightforward if we consider $f$ and $g$ close in $\mathcal C^1(K)$-norm instead of the sup-norm on $K$? Because this way we would have both $f-g$ and $f'-g'$ small on $K$. So it is rather clear that for $\eta>0$ small enough, we would achieve the two properties above for $g$ on some compact $\widetilde K\subset K^{\circ}$.
Is this correct?