Category Archives: Sequence

Rudin- Ch. 3 #13

Problem Statement: Given and define the product to be where . Suppose that converges to absolutely and converges to absolutely. Prove that converges to a value absolutely. Proof: Since and converge absolutely we know that their product converges to (this is by … Continue reading

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Rudin: Ch. 3 #8

Problem Statement: If converges and is monotonic and bounded, prove that converges. Proof: Since we are given that is bounded and monotonic we know there exists some such that . Let , then there exists some such that for every . Note that … Continue reading

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May 1999 #1

Problem Statement: Let be a sequence of real numbers. Prove that converges if and only if converges. Proof:  Assume that converges to some real number . Let . Then there exists an such that for every . Consider . . Let , … Continue reading

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Winter 1995

Problem Statement: Let and let for all and for every . Show that if converges then converges uniformly on . Note: This is asking us to prove the Weierstrass M-Test. Proof: Let and for every and for every . Assume that converges. … Continue reading

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Winter 1995 #4

Problem Statement: Let be uniformly continuous and let for every . Show converges uniformly. Proof: First I claim that converges pointwise to . Let . Then . Since is continuous and converges to it follows that . This holds for all … Continue reading

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Winter 1995 #1

Problem Statement: Using only the definition of a Cauchy sequence show that every Cauchy sequence in is bounded. Proof: Let be a Cauchy sequence in . Let , then there exists a such that for every it follows that . Let latex … Continue reading

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Cauchy Condensation Test

The Cauchy Condensation Test states that for a decreasing sequence where for all then converges if and only if converges. Proof: In the forward direction, let us assume that converges. This means that the sequence of partial sums, converge. Define … Continue reading

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