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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. LetS⊆Rbe non-empty and bounded above and letu= supS. Prove thatu∈Sif andonly ifu= maxS.4. (a) LetS⊆Rbe non-empty and bounded above and letu= supS. Prove thatuis unique.(b) Prove that if each ofmandnis a maximum ofS,thenm=n.5. LetS⊆Rand suppose thatv= infS. Prove that for any positive number,there is anelements∈Ssuch thatv≤s< v+ .6. Prove that ifxandyare real numbers withx < y,then there are infinitely rationalnumbers in the interval[x, y].
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I.4.
TOPOLOGY OF THE REALS
Definition 6.LetS⊆R. The setSc={x∈R:x /∈S}is called thecomplementofS.Definition 7.Letx∈Rand let>0.An-neighborhoodofx(often shortened to“neighborhood ofx”) is the setN(x,) ={y∈R:|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.Letx∈Rand let>0. Adeleted-neighborhoodofx(often shortened to“deleted neighborhood ofx”) is the setN*(x,) ={y∈R: 0<|y-x|<}.A deleted-neighborhood ofxis an-neighborhood ofxwith the pointxremoved;N*(x,) = (x-, x)∪(x, x+).

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- Fall '08
- Staff
- Topology, Real Numbers, Natural Numbers, Empty set, Metric space, Closed set, Nxk