math-341.pdf - MATH 341 Vincent Jodoin Definitions Axiom of...

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MATH 341Vincent JodoinApril 17, 2018Definitions
Def 1.3.1 - A setARis bounded above if there exists a numberbRsuch thatabfor allaA.The numberbis called an upper boundforA. Similarly, the setAis bounded below if there exists a lower boundlRsatisfyinglafor everyaA.Def 1.3.2 - A real numbersis the least upper bound for a setARif itmeets the following two criteria: (i)sis an upper bound forA; (ii) ifbisany upper bound forA, thensb.Def 1.3.4 - A real numbera0is a maximum of the setAifa0is an elementofAanda0afor allaA. Similarly, a numbera1is a minimum ofAifa1Aanda1afor everyaA.
Anis countable for eachnNthenS1Anis countable.Def 2.2.1 - A sequence is a function whose domain isN.Def 2.2.3 (Convergence of a Sequence)- A sequence(an)converges to areal numberaif, for every positive numberε, there exists anNNsuchthat whenevernNit follows that|an-a|< ε.1
Def 2.2.3B (Convergence of a Sequence: Topological Version) - A sequence(an)converges to a if, given anyε-neighborhood Vε(a)ofa, there existsa point in the sequence after which all of the terms are inVε(a). In otherwords, everyε-neighborhoodcontains all but a finite number of the termsof(an).Def 2.2.4 - Given a real numberaRand a positive numberε >0, thesetVε(a) ={xR:|x-a|< ε}is called theε-neighborhoodofa.

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