Existence of net of positive real numbers

Question

Does for every totally ordered infinite set $T$ with no greatest element, there exist a net $(a_t )_{t\in T}$ of positive real numbers which is convergent to $0.$ ?

Answer 1

It may not be possible. Consider the least uncountable ordinal $\omega_1 = \{ \alpha : \alpha < \omega_1 \}$. This set is totally ordered by $<$ and has no upper bound. It is known that there can be no strictly monotone net of real numbers indexed by $\omega_1$ (If $\langle a_\alpha \rangle_{\alpha < \omega_1}$ were strictly increasing, then for each $\alpha < \omega_1$ pick a rational number $q_\alpha$ with $a_\alpha < q_\alpha < a_{\alpha+1}$. How many $q_\alpha$s have we chosen?)

If $\langle a_\alpha \rangle_{\alpha < \omega_1}$ were a net of positive real numbers converging to $0$ it follows that there is an uncountable (and cofinal) $A \subseteq \omega_1$ such that the sub-net $\langle a_\alpha \rangle_{\alpha \in A}$ is strictly decreasing (and converges to $0$). Now modify an observation given above to get a contradiction.