A common strategy to prove things is to use null sequences and I was wondering if it is possible to generalise the concept to aritrary nets.
Let $I$ be a directed set. Is there a net $\left( x_\alpha \right)_{\alpha \in I}$ of positive real numbers such that $lim_\alpha x_\alpha = 0$? This is certainly equivalent to the existence of a monotonic and final map from $I$ to $\mathbb{R}$ and perhaps even to $\mathbb{N}$. My gut feeling tells me that such maps do not exist for "too large" $I$, i.e that there are totally ordered sets $I$ for which such monotonic maps must eventually be constant.
But my intuition might be tainted because I would attempt to construct such a map by taking a strictly monotonic and cofinal map from $\mathbb{N}$ to $I$ (which most certainly does not always exist) and then to "fill the gaps" (which also does not work unless $I$ is totally ordered).
Thus, I could imagine that there are non-constructive ways to tackle this problem by starting with some arbitrary surjective map from $I$ to $\mathbb{N}$.
Quite simply, let $I=(\omega_1,\le)$ be the first uncountable ordinal, let $(x_i)_{i\in I}$ be a null net and call $\alpha_n=\min\{i\in I\,:\, \forall j\ge i, x_j\le1/n\}$. Since $\operatorname{cof}\omega_1=\omega_1>\omega$, we have that $\beta=\sup_{n\in\Bbb N}\alpha_n<\omega_1$. By definition, $x_\gamma$ cannot be positive for any $\gamma\ge\beta$.