# Category Archives: Cauchy

## 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

## May 1999 #5

Problem Statement: Let be a real valued differentiable function on whose derivative is bounded on .  Show that converges. Proof: First note that for every . Furthermore we know that is bounded so there exists a real number such that for every … Continue reading

## 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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## September 1999 #1

Problem Statement: Let be a sequence of non-negative real numbers. Assume there exists a real number such that converges. Show converges. Proof: Let be a sequence of non-negative real numbers. We know there exists a real number such that converges. Since the … Continue reading

## September 2001 #2

Problem Statement: Suppose that converges absolutely. Prove that converges for every . Proof: We are given that  converges absolutely, this implies that  converges. By definition this means that the sequence of partial sums, , converges. But every convergent sequence is also Cauchy. Let … Continue reading

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## September 2001 #1

Problem Statement: Let be differentiable on and suppose that exists and is finite. Prove if exists and is finite, then . Proof: Suppose that and that . Since these limits exist we know that these limits will exist, and be the … Continue reading