The limsup on sequences of extended real numbers is usually taken to be either of these two things, which are equivalent:
- the sup of all subsequential limits.
- The limit of the sup of the tail ends of the sequence.
For the situation with nets, the same arguments guarantee the existence of the above quantities 1. and 2, as long as you understand that a subnet of a net must be precomposed with an increasing function that is also cofinal. Also, one has that 1. $\leq$ 2., and I just can't see the reverse inequality. Don't forget that one cannot imitate the argument for sequences because the following fact fails:
Given a directed set, how do I construct a net that converges to $0$.
Best Answer
As far as I can say, the more usual definition of limit superior of a net is the one using limit of suprema of tails: $$\limsup x_d = \lim_{d\in D} \sup_{e\ge d} x_e = \inf_{d\in D} \sup_{e\ge d} x_e.$$
But you would get an equivalent definition, if you defined $\limsup x_d$ as the largest cluster point of the net. This definition corresponds (in a sense) to the definition with subsequential limits, since a real number is a cluster point of a net if and only if there is a subnet converging to this number.
I think that it is relatively easy to see that $\limsup x_d$ is a cluster point of the net $(x_d)_{d\in D}$.
To see that for every cluster point $x$ we have $x\le\limsup x_d$ it suffices to notice that, for any given $\varepsilon>0$ and $d\in D$, the interval $(x-\varepsilon,x+\varepsilon)$ must contain some element $x_e$ for $e\ge d$. Hence we get $$ \begin{align*} x-\varepsilon &\le \sup_{e\ge d} x_e\\ x-\varepsilon &\le \lim_{d\in D} \sup_{e\ge d} x_e. \end{align*}$$ and, since $\varepsilon>0$ is arbitrary, we get $$x\le \lim \sup_{e\ge d} x_e.$$
Thus the limit superior is indeed the maximal cluster point.
So the only thing missing is to show that cluster points are precisely the limits of subnets - this is a standard result, which you can find in many textbooks.
Some references for limit superior of a net are given in the Wikipedia article and in my answer here.
Perhaps some details given in my notes here can be useful, too. (The notes are still unfinished.) I should mention, that I pay more attention there to the notion of limit superior along a filter (you can find this in literature defined for filter base, which leads basically to the same thing). The limit superior of a net can be considered a special case, if we use the section filter; which is the filter generated by the base $\mathcal B(D)=\{D_a; a\in D\}$, where $D_a$ is the upper section $D_a=\{d\in D; d\ge a\}$.