Math 4661/6661 Quiz 4 12/3/2012 Name: 1. Let D(a, b; r) denote the open disc in R2 with center at (a, b) and radius r. Determine if the set B = {D(a, b; 1 n):(a, b) ? Q2, n ? N} is countable. Justify your answer. 2. Given that a sequence (bn) of real numbers does not converge to ß. Prove that (bn) has a subsequence that has no subsequence converging to ß. 3. Find the upper limit of the sequence (an) = (1 + sin n + cos2 n). 4. Let !8 n=1 an be a convergent series of positive terms and let rn = !8 k=n ak, for each n. (a). Show that the series !8 n=1 v an rn converges. (b). Show that the series !8 n=1 an rn diverges. 5. Suppose that the series !8 n=1 an converges and the sequence (cn) is bounded and decreasing. Show that the series !8 n=1 ancn converges. Hint: Let c = limn?8 cn and bn = cn-c. Apply Abel’s Test to !8 n=1 anbn. Note that the sum of two convergent series is convergent. 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. Hint: Assume otherwise. Explain why for each n ? N, there exist xn ? K and yn ? Uc such that |xn - yn| Document Preview: 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 |

Math 4661/6661 Quiz 4 12/3/2012 Name: 1. Let D(a, b; r) denote the open disc in R2 with center at (a, b) and radius r. Determine if the set B = {D(a, b; 1 n):(a, b) ? Q2, n ? N} is countable. Justify...

Math 4661/6661 Quiz 4 12/3/2012 Name: 1. Let D(a, b; r) denote the open disc in R2 with center at (a, b) and radius r. Determine if the set B = {D(a, b; 1 n):(a, b) ? Q2, n ? N} is countable. Justify your answer. 2. Given that a sequence (bn) of real numbers does not converge to ß. Prove that (bn) has a subsequence that has no subsequence converging to ß. 3. Find the upper limit of the sequence (an) = (1 + sin n + cos2 n). 4. Let !8 n=1 an be a convergent series of positive terms and let rn = !8 k=n ak, for each n. (a). Show that the series !8 n=1 v an rn converges. (b). Show that the series !8 n=1 an rn diverges. 5. Suppose that the series !8 n=1 an converges and the sequence (cn) is bounded and decreasing. Show that the series !8 n=1 ancn converges. Hint: Let c = limn?8 cn and bn = cn-c. Apply Abel’s Test to !8 n=1 anbn. Note that the sum of two convergent series is convergent. 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. Hint: Assume otherwise. Explain why for each n ? N, there exist xn ? K and yn ? Uc such that |xn - yn| < 1/n.="" note="" that="" k="" is="" compact,="" uc="" is="" closed="" and="" k="" n="" uc="">

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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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Answer To: Math 4661/6661 Quiz 4 12/3/2012 Name: 1. Let D(a, b; r) denote the open disc in R2 with center at...

Robert answered on Dec 23 2021

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1
1. The Cartesian product of finitely many countable sets is countable. Moreover your D(a, b, 1n ) where (a, b) ∈ Q
2 is
will lie in the cartesian product of Q×Q× Z which is countable.
Solution. 2 Suppose {bn} has a subsequence {bnk} which converges to β. Then by definition we have that there is
a N ∈ N such that for nk > N we have |bnk − β| <...

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Math 4661/6661 Quiz 4 12/3/2012 Name: 1. Let D(a, b; r) denote the open disc in R2 with center at (a, b) and radius r. Determine if the set B = {D(a, b; 1 n):(a, b) ? Q2, n ? N} is countable. Justify...

Answer To: Math 4661/6661 Quiz 4 12/3/2012 Name: 1. Let D(a, b; r) denote the open disc in R2 with center at...

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