Show an example of an uniform convergent sequence of injective functions s.t. the limit function is not injective.
My approach would be to consider \begin{align*} &f_n:D\rightarrow \mathbb{C}\\ &f_n(z)=\frac{z}{n} \end{align*}
which is injective. $(f_n)_{n\in\mathbb{N}}$ is an uniform convergent sequence (with $f(z) = 0$ $\forall z$), because for $|z|\leq R$ we have: $$|f_n(z) -f(z)| = |\frac{z}{n}-0| = \frac{|z|}{n}\leq\frac{R}{n} \overset{n\rightarrow \infty}{\longrightarrow} 0$$
So the sequence is uniform convergent for any closed disk $D$ with radius $R$.
Since $f_n \rightarrow f = 0$ and we know that constant functions are clearly not injective, we found such an example.
Is that correct or did I miss something? The idea for this example comes from a theorem I learned in lecture which states...
If $U$ is a domain, $f_n:U\rightarrow \mathbb{C}$ injective and $(f_n)_{n\in\mathbb{N}}$ uniform convergent to $f$, then $f$ is either injective or constant