Math 4661/6661 Quiz 4 12/3/2012 Name: 2 1. Let D(a,b;r)denotetheopendiscinR with center at (a,b)andradius r.Determineifthe 1 2 set B = {D(a,b; ):(a,b)?Q ,n?N} is countable. Justify your answer. n 2. Given that a sequence (b)ofrealnumbersdoesnotconvergeto ß.Provethat(b)hasa n n subsequence that has no subsequence converging to ß.2 3. Find the upper limit of the sequence (a )=(1+sinn+cos n). n ! ! 8 8 4. Let a be a convergent series of positive terms and let r = a,foreach n. n n k n=1 k=n ! 8 a n v (a). Show that the series converges. n=1 r n ! 8 a n (b). Show that the series diverges. n=1 rn! 8 5. Suppose that the series a converges and the sequence (c )isboundedanddecreasing. n n n=1 ! 8 Show that the series a c converges. Hint: Letc=lim c andb =c -c.ApplyAbel’s n n n?8 n n n n=1 ! 8 Test to a b .Notethatthesumoftwoconvergentseriesisconvergent. n n n=1 6. Let K ?R be a nonempty compact set and let U ?R be an open set containing K.Prove that there is a positive number " such that for every x?K,N (x)=(x-",x+")?U. ! c Hint: Assume otherwise. Explain why for each n?N,thereexist x ? K and y ? U such that n n c c |x -y |<>
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