Let $\mathscr{T}$ be the set of partitions $\tau = (\tau_0 = 0 < \tau_1 < \dots < \tau_N = 1)$ of the interval $[0,1]$ (where $N$ is not fixed). This becomes a directed set by setting $\tau < \tau^\prime$ iff $\tau^\prime$ is a subdivision.
Now we can look at the nets $$ s_\tau := \sum_{j=1}^N (\tau_j - \tau_{j-1})^2$$ or $$ r_\tau := N \sum_{j=1}^N (\tau_j - \tau_{j-1})^3$$
We should have both $s_\tau \longrightarrow 0$ and $r_\tau \longrightarrow 0$ in the sense of nets, but I don't know how to prove it. Does anybody have an idea?
A general hint and a partial illustration.
As I understood, you must show that for each $\varepsilon>0$ there are nets $\tau_1$ and $\tau_2$ such that $s_{\tau_1’}<\varepsilon$ and $r_{\tau_2’}<\varepsilon$ for each $\tau_1’>\tau_1$ and $\tau_2’>\tau_2$.
Since $(a+b)^2\ge a^2+b^2$, provided $a$ and $b$ are non-negative, $s_\tau$ is monotone. So it is enough to show that for each real $\varepsilon>0$ there is a net $\tau$ such that $s_\tau<\varepsilon$.