# N then x n y n is cauchy the sets n and z have the

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n, then{xn/yn}is Cauchy.The setsNandZhave the same cardinality.
5.(14 marks)LetEbe a subset ofR. A pointaRis called a cluster point ofEifE(a-r, a+r) contains infinitelymany points for anyr >0. Prove thatais a cluster point ofEif and only if for eachr >0,E(a-r, a+r)\{a}is nonempty.Prove that every bounded infinite subset ofRhas at least one cluster point.
6.(10 marks)Assume thatf: ([a, b][c, d])Ris a continuous function, wherea < b < c < dare real numbers.We also know that [c, d]f([a, b]) and [a, b]f([c, d]). Prove that there exists somexwithaxbsuch thatf(f(x)) =x.
7. EXTRA PROBLEMS FOR EXTRA MARKS(5 marks)Assume thatA, BandCare subsets ofNsuch thatABC=Nand all the setsAB,AC,BCare infinite. Is it possible thatABC=?(10 marks)Prove the second part of de Morgan’s law, i.e. the following statement.IfE={Eα}αAwithEαXand the we writeEc=X\E, then(αAEα)c=αAEcα.(12 marks)Prove that limn→∞an=ARimplies limN→∞1NNn=1an=A. (Hint: try to chop the suminto two parts.)