Countable and net convergence

Marsden, J. E.

Amer. Math. Monthly, 75, (1968)

Abstract:

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, $\lim\limits_{{x \to y}}^{}$f (x) = a   iff$\lim\limits_{{n\to \infty}}^{}$f (x_n) = a for every sequence x_n$\to$y . The purpose of this note is to isolate the basic reasons for these phenomena.