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Similar to b x for each x belonging to enjoyable the inequality the set Z, we name the set Z bounded above. equally, a collection Z is named bounded less than if a bunch exists < x. pleasurable the above situation: m m The geometrical that means of those principles is the subsequent. is bounded it signifies that it truly is contained in a undeniable section at the numerical line. a suite is bounded above or less than, whilst it really is contained in an unlimited radius If a suite directed to the left or to the precise, respectively. utilising the continuity precept we shall turn out the following theorem: Z // a non-empty set numbers M, pleasurable is the belonging to Z, there exists be known as the top certain Z bounded above, then one of the > x for any x inequality a least one. This quantity will M of the set Z. bounded under, then one of the numbers m, pleasing the inequality m < x for any x belonging to Z, there exists a greatest^one, called the decrease certain of the set Z. equally, if a set facts. allow the set Z is be bounded above. allow us to divide the set of all actual numbers in periods as follows. To the second one category there'll belong all numbers > x for any x belonging to Z. gratifying the inequality M M To the firstclass there'll belong all different actual numbers; quantity a belongs to the 1st category that implies undefined while within the set one of these department is known as a minimize, Z there exists a of the set of actual i. e. quantity more than a. numbers in periods any quantity belonging to the 1st than any quantity belonging to the second. of the second one type to be certainly, supposing a bunch under a host a of the 1st classification and realizing quantity x exists within the set Z such < x we should always have < x; yet this contradicts the definition of the category is much less M M moment type. in addition, allow us to observe that either sessions are non-empty. certainly, if a bunch z belongs to the set Z (and such 1. creation 21 a bunch exists, the set Z being non-empty by means of assumption), then the quantity belongs to the 1st type. the second one classification is usually non-empty, because the set Z is bounded z\ above. in response to the continuity precept there exists both a maximum quantity within the firstclass or a least quantity within the moment category. even if, the 1st eventuality impossible. particularly, if a belongs to the 1st type and a < x (where x belongs to Z), then denoting via a' is any quantity among a and e. g. a? , a' = T* / feet I , we now have 2i < yet which means a' belongs to the 1st a of the 1st type we type, too. hence, to any quantity may well discover a quantity a' more than a during this type.