See if you could solve this, /sci/, as an exercise. It's not homework, don't worry. I already have my solution I will post afterwards.

Let (M, J) be any topological space. For generalized sequences (nets) (Fsubi) [for all i in I] of subsets of M one defines the limes inferior of Fsubi and limes superior of Fsubi and convergence analogously as for sequences. Show that every generalized sequence of subsets of M contains a converging generalized subsequence. (Hint: use Tychonoff's Theorem)

Penguins for motivation.