Editing Monotone convergence theorem (section)

==Convergence of a monotone series==

There is a variant of the proposition above where we allow unbounded sequences in the extended real numbers, the real numbers with <math>\infty</math> and <math> -\infty</math> added. 
:<math> \bar\R = \R \cup \{-\infty, \infty\}</math>
In the extended real numbers every set has a [[supremum]] (resp. [[infimum]]) which of course may be <math>\infty</math> (resp. <math>-\infty</math>) if the set is unbounded. An important use of the extended reals is that any set of non negative numbers <math> a_i \ge 0, i \in I </math> has a well defined summation order independent sum 
:<math> \sum_{i \in I} a_i = \sup_{J \subset I,\ |J|< \infty} \sum_{j \in J} a_j \in \bar \R_{\ge 0}</math>
where <math>\bar\R_{\ge 0} = [0, \infty] \subset \bar \R</math> are the upper extended non negative real numbers. For a series of non negative numbers 
:<math>\sum_{i = 1}^\infty a_i = \lim_{k \to \infty} \sum_{i = 1}^k a_i = \sup_k \sum_{i =1}^k a_i = \sup_{J \subset \N, |J| < \infty} \sum_{j \in J} a_j = \sum_{i \in \N} a_i,</math>
so this sum coincides with the sum of a series if both are defined. In particular the sum of a series of non negative numbers does not depend on the order of summation.