I'm still unfamiliar with nets and I'm trying to get some intuitions about them right. Here is one question I have.
Question Let $\{x_\alpha\}\subseteq\mathbb C$ be a convergent net converging to $x\in \mathbb C$. If $|x_\alpha|\leq 1$ for all $\alpha$, then $|x|\leq 1$.
Attempt Suppose not, i.e., $|x|>1$. Pick an $r>0$ small enough so that the ball $B(x,r)$ centered at $x$ with radius $r$ does not intersect the unit disk $\mathbb D$. Since $B(x,r)$ is open, there exists $\alpha_0$ such that for all $\beta\geq \alpha_0$ $x_\beta\in B(x,r)\subseteq \mathbb C\setminus \mathbb D$. In other words, $|x_\beta|>1$, which is a contradiction.
Is this correct or there is a subtle flaw somewhere?
Your argument is indeed correct. It can be used to something more general: if $C$ is closed and $x_\alpha$ is a net from $C$ converging to $x$ then $x \in C$. This can be applied to the closed set $C=\{z \in \Bbb C: |z| \le 1\}$.