midterm1-sol.pdf - Analysis Math 325 Midterm 1 October 1...

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Analysis, Math 325Midterm 1October 1, 20171. Letxn=9n+ 93n+ 1.Arguing directly from the definition of a limit, prove that{xn}is convergent and find its limit.
2.LetAandBbe nonempty subsets ofRthat are bounded from below and leta=infAandb=infB.Moreover, assume thatABis not empty. Show thatinfABmax{a, b}. Provide an example whenthe inequality is strict.
3.Suppose that{xn}converges toxand{yn}converges toy, and suppose thatx < y. Prove the followingstatement: There existsNNsuch that for allnN,xn< yn.

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