The Density of the Rational Numbers and the Irrational Numbers Lemma Let y be a

The density of the rational numbers and the

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Ifxandyare real numbers,x < y,then there exists an irrational numberzsuch thatx < z < y. Exercises 1.3 1. True – False.Justify your answer by citing a theorem, giving a proof, or giving a counter-example. 2. True – False.Justify your answer by citing a theorem, giving a proof, or giving a counter-example. 3. LetSRbe non-empty and bounded above and letu= supS. Prove thatuSif andonly ifu= maxS.4. (a) LetSRbe non-empty and bounded above and letu= supS. Prove thatuis unique.(b) Prove that if each ofmandnis a maximum ofS,thenm=n.5. LetSRand suppose thatv= infS. Prove that for any positive number,there is anelementsSsuch thatvs< v+ .6. Prove that ifxandyare real numbers withx < y,then there are infinitely rationalnumbers in the interval[x, y]. 7
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I.4. TOPOLOGY OF THE REALS Definition 6.LetSR. The setSc={xR:x /S}is called thecomplementofS.Definition 7.LetxRand let>0.An-neighborhoodofx(often shortened to“neighborhood ofx”) is the setN(x,) ={yR:|y-x|<}.The numberis called the radiusofN(x,).Note that an-neighborhood of a pointxis the open interval(x-, x+)centered atxwith radius.Definition 8.LetxRand let>0. Adeleted-neighborhoodofx(often shortened to“deleted neighborhood ofx”) is the setN*(x,) ={yR: 0<|y-x|<}.A deleted-neighborhood ofxis an-neighborhood ofxwith the pointxremoved;N*(x,) = (x-, x)(x, x+).
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