Editing Net (mathematics) (section)

===Limit superior/inferior===

[[Limit superior]] and [[limit inferior]] of a net of real numbers can be defined in a similar manner as for sequences.<ref>Aliprantis-Border, p. 32</ref><ref>Megginson, p. 217, p. 221, Exercises 2.53–2.55</ref><ref>Beer, p. 2</ref> Some authors work even with more general structures than the real line, like complete lattices.<ref>Schechter, Sections 7.43–7.47</ref>

For a net <math>\left(x_a\right)_{a \in A},</math> put
<math display=block>\limsup x_a = \lim_{a \in A} \sup_{b \succeq a} x_b = \inf_{a \in A} \sup_{b \succeq a} x_b.</math>

Limit superior of a net of real numbers has many properties analogous to the case of sequences. For example,
<math display=block>\limsup (x_a + y_a) \leq \limsup x_a + \limsup y_a,</math>
where equality holds whenever one of the nets is convergent.