B x a b is not a complete subset of r for any a b r

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(b)X= (a, b) is not a complete subset ofRfor anya, bRwitha < b. [4 marks]
(c)Zis a complete subset ofR. [5 marks]
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Trivially, this implies that (an)n=1converges toaNX, soXis complete.S5. Determine whether each of the following statements is true or false. Briefly justify eachanswer; for false statements, providing a counterexample is sufficient. For this question,(an)n=1is a sequence of real numbers. [3 marks each](a) If (an)n=1converges, then the seriesn=1anconverges.
(b) If a seriesn=1anconverges, then the sequence (an)n=1converges.
(c) If the sequence of partial sums (sn)n=1=nXk=1akis Cauchy, then the seriesn=1anconverges.
(d) If given anyε >0, there exists an integerNsuch thatXk=n+1ak< εfor allnN,then the seriesn=1anconverges.Xk=n+1ak< εmeans that the series (fromn+ 1
S6. Let (an)n=1be a sequence such that limn→∞|an|= 0. Show that there exists a subsequence(ank)k=1such that the seriesXk=1ankconverges. [10 marks]
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