Supremum_and_Infimum

Supremum_and_Infimum 1 Supremum and Infimum These are like maximum and minimums for the case of infinite sets S. Supremum: Let’s start by defining S as a subset of the real line,  . A real number,  , is said to be an upper bound of S if x for every S...

1 Supremum and Infimum These are like maximum and minimums for the case of infinite sets S. Supremum: Let’s start by defining S as a subset of the real line,  . A real number,  , is said to be an upper bound of S if x for every Sx , and no number smaller than  is an upper bound of S. Thus, no member of the set exceeds  . But if 0 (however small), there is a member of the set that exceeds   . The supremum of S (denoted as sup S) is the least element of  that is greater than or equal to all elements of S. The Supremum Property is: If a set S has a supermum in  , then for any 0 , there is a Sx such that SxS supsup  This is sometimes referred to as the least upper bound as well as minimal upper bound or maximal element. Notice that if  is an upper bound of S is the supremum of S then any number larger than  is an upper bound of S, and any number smaller than  is not an upper bound. Hence, S must contain numbers that are arbitrarily close to  . The definitions of upper bound and supremum of a set S do not require that these numbers belong to S. If S has a supremum  and  is an element of S, we call  the maximum of S, denoted as Smax . Example: The interval (0,1] has a maximum equal to 1, whereas (0,1) has no maximum, although it does have a supremum equal to 1. Infimum: Now consider a real number  which is a lower bound of S if x for every Sx , and no number greater than  is an lower bound of S. 2 Thus, no member of the set is less than  . But if 0 (however small), there is a member of the set that is less than   . The infimum of S (denoted as inf S) is the greatest element of  that is less than or equal to all elements of S. The Infimum Property is: If a set S has an infimum in  , then for any 0 , there is a Sx such that  SxS infinf This is sometimes referred to as the greatest bound as well as maximal lower bound or minimal element. Notice that if  is a lower bound of S is the infimum of S then any number smaller than  is an lower bound of S, and any number greater than  is not an lower bound. Hence, S must contain numbers that are arbitrarily close to  . Like the case of the supremum, the definitions of lower bound and infimum of a set S do not require that these numbers belong to S. If S has a infimum  and  is an element of S, we call  the minimum of S, denoted as Smin . Example: We can reverse the earlier example to be The interval (0,1] has a maximum equal to 1, whereas (0,1) has no maximum, although it does have a supremum equal to 1. The interval [0,1) has a minimum equal to 0, whereas (0,1) has no minimum, although it does have a infimum equal to 0.

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Supremum_and_Infimum

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