So i made a convergence argument using nets where i used the triangle inequality.
For a simplified example of what im talking about lets say i have something like a net $(a)_{i\in I}$ in $\mathbb{R}$ with$$|a - a_i| \leq \epsilon~~\forall i\geq i_1$$ and $$|b-a_i| \leq \epsilon~~\forall i \geq i_2$$ can i write something like $$|b-a| \leq |b-a_i+a_i-a| \leq |b-a_i| + |a - a_i| \leq 2 \epsilon~\forall i \geq \max(i_1,i_2)$$ or is the expression $\max(i_1,i_2)$ nonsense for elements of a general directed set? if it happens to be nonsense how do i write such a situation up correctly? does $\sup(i_1,i_2)$ work instead?
i know directed sets posess upper bounds for finite subsets, but the two elements need not be comparable ... so maybe i'm supposed to write something like "this inequality holds for all $i$ greater than some common upper bound of $i_1 \text{ and }i_2 $" instead? whats a standard notion for this?
thanks in advance!
You don't always have a max, but a directed set always has a common upper bound for two elements ( not necessarily a minimal upper bound, which would be a sup): so you can write "there exists $i_3 \in I$ so that $i_3 \ge i_1$ and $i_3 \ge i_2$" and then any $i \ge i_3$ will obey both conditions (by transivity: $i \ge i_3 \ge i_1 \to i \ge i_1$ etc.).
That's why nets are defined on directed sets, not posets, say. The common upperbound is the "direction" part, as it were.
AFAIK there is no standard shorthand or notation for this. You can just appeal to the existence of such a common upper bound $i_3$.
In many common cases there will be a sup but you cannot rely on it in a general argument on nets.