Transcription of Chapter2
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Chapter2
Chapter 2 The 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. Ifm Ris a lower bound ofAsuch thatm m for every lower boundm ofA, thenmis called the or infimumofA, denotedm= not bounded from above , then we write supA= , and ifAis notbounded from below, we write infA=.
from above, so sup(A+B) exists if and only if both supA and supB exist. In that case, if x ∈ A and y ∈ B, then x+y ≤ supA+supB, so supA +supB is an upper bound of A +B and therefore sup(A +B) ≤ supA+supB. To get the inequality in the opposite direction, suppose that ǫ > 0. Then there
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