Let $(v_\alpha)_{\alpha \in I}$ be a net in $\mathbb{C}^m$ such that $v_\alpha \not\to 0$. Show that there is a subnet $(w_\beta)_{\beta \in J}$ of this net with the property that $w_\beta \neq 0$ for all $\beta \in J$ and such that there is $\delta \geq 0$ such that $$\frac{1}{\Vert w_\beta\Vert} \to \delta $$
How can I show this? From $v_\alpha \not \to 0$, I know that
$$\exists \epsilon > 0: \forall \alpha \in I: \exists \alpha_0 \geq \alpha: \Vert v_{\alpha_0} \Vert \geq \epsilon$$
and using this I should be able to construct the given subnet.
How can I proceed?
Fix that $\epsilon>0$ and let $J$ be the subset of $I$ of all $j\in I$ such that $\lVert v_j\rVert\geq\epsilon$. Your condition simply means that $J$ is cofinal with $I$.
Therfore $s:J\to\mathbb{C}^m$ given by $s_j=v_j$ is a subnet of $v$. Additionally $\lVert s_j\rVert\geq \epsilon$ for all $j\in J$ and thus $\frac{1}{\lVert s_j\rVert}$, which is a net $J\to\mathbb{R}$, is bounded. I.e. the image of $s$ is fully contained in some closed interval $D\subseteq\mathbb{R}$ which is compact. Therefore it has a convergent subnet, say $t:T\to\mathbb{R}$, by the net characterization of compact spaces. This subnet $t$ comes together with a monotone, cofinal function $F:T\to J$, by the definition of subnet. You can easily check that $s\circ F$ is the subnet of $v$ you were looking for.