Note that in a directed set every finite subset has an upper bound. If the directed set itself is finite, then it must have a maximum (why?). Therefore the limit of such net is the element associated with the maximum index.

Convergence is trivial in that case.

Can directed set be finite?
In OpenTTD, it's possible to build multi-platform train stations side-by-side (pictured below as Parnpool Mines), and they would work as expected, allowing two trains to load/unload at that station at the same time. 

However, sometimes due to space constraints it would be more desirable to to build the two platforms lengthwise as shown below in Parnpool West. The signal-station combination pictured below, however, does not work, and I have not found one that does.


