Running through some geometry papers, I found some authors use the following idea:
Let $f_n : \Bbb C^m \to \Bbb C^m$ be a sequence of holomorphic functions converging uniformly to $f : \Bbb C^m \to \Bbb C^m$ on compact sets. Suppose $f(0) =0$ and $\frac{\partial f }{\partial z}$ is invertible (ie, the inverse function theorem can be applied). Then for any $\varepsilon >0$ there is some $N$ and a point $z \in \Bbb C^m$ such that $f_N(z)=0$ and $|z|< \varepsilon$.
I know the result is true for $m=1$, it just comes as a consequence of the argument principle, but I of course I cannot generalize to more dimensions. On top of that I have seen the problem for $m=1$ in some complex analysis books or lecture notes, which makes me think that the result exchanging $\Bbb C$ by $\Bbb R$ (and holomorphic by smooth or analytic) may not be true, which is problematic because it means that the theory of several complex variables is needed, something I am not very familiar with.
Is the result true? References or direct proofs? Are there related results? Is is true for $\Bbb R $ instead of $\Bbb C$?
I'd say maximum modulus principle:
For $\epsilon>0$ fixed small enough, such that $|f|$ is non-zero on $|z|=\epsilon$ and $\det(\nabla f(z))$ is non-zero on $|z|\le \epsilon$.
For $N$ large enough and $|z|<\epsilon$,
$\det(\nabla f_N(z))$ is non-zero (being close to $\det(\nabla f(z))$).
(this was the only steps where I'm using that the functions are holomorphic, so that the derivatives are continuous and converge uniformly)
If $f_N(z)\ne 0$ then there is a direction $v=v(z,N)$ such that the directional derivative is $D_vf_N(z)= -f_N(z)$ ie. $f_N(z+hv) = f_N(z)-h f_N(z)+O(h^2)$ and thus $z$ is not a local minimum of $|f_N|$.
$\inf_{|z|\le \epsilon} |f_N(z)|$ can't be on $|z|=\epsilon$ since $|f_N|$ converges uniformly to $|f|$.
Whence there is some $|z|<\epsilon$ where $f_N(z)=0$.