9.1 Useful Results in Analysis

We first recall some elementary notation used in the book. For any real number x, x ⁺ = max{ x, 0} and x ^? = max{ ? x, 0} so x = x ⁺ ? x ^?. When we write X = X ⁺ ? X ^? for a random variable X we simply mean that the functions are defined for each ?; i.e. X( ?) = X( ?) ⁺ ? X( ?) ^?. We also denote the infimum and supremum of a sequence of real numbers with index set by inf and sup respectively. The supremum is the least upper bound of the set of x's while the infimum is the greatest lower bound. If U is the supremum then x _n, ? U for all n and for any ? no matter how small we can find an m such that x _m ? U ? ?. Similarly, if L is the infimum then x _n ? L for all n and for any ? no matter how small we can find an m such that x _m ? L + ?.

The limit of a sequence { x _n, n ? {1, 2, } is denoted by lim x _n ?