Chapter2

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Chapter 2The supremum and infimumWe review the definition of the supremum and and infimum and some of theirproperties that we use in defining and analyzing the Riemann DefinitionFirst, we define upper and lower setA Rof real numbers is bounded from above if there existsa real numberM R, called an upper bound ofA, such thatx Mfor everyx A. Similarly,Ais bounded from below if there existsm R, called a lowerbound ofA, such thatx mfor everyx A. A set is bounded if it is boundedboth from above and supremum of a set is its least upper bound and the infimum is its greatestupper thatA Ris a set of real numbers. IfM Ris anupper bound ofAsuch thatM M for every upper boundM ofA, thenMiscalled the supremum ofA, denotedM= supA.

A function f is bounded from above on A if supAf is finite, bounded from below on A if infAf is finite, and bounded on A if both are finite. Inequalities and operations on functions are defined pointwise as usual; for example, if f,g : A → R, then f ≤ g …

  Bounded