Countable and net convergence
Marsden, J. E.
Amer. Math. Monthly, 75, (1968)
It is well known that Lebesgue's dominated convergence theorem does not hold for nets; that is, having a countable sequence is essential. On the other hand, for a real valued function on an interval, sequences do suffice; that is,
f (x) = a ifff (xn) = a
for every sequence xny
. The purpose of this note is to isolate the basic reasons for these phenomena.