Tag Archives: mathematics

September 2001 #5

Problem Statement: Use the open cover definition of compact to show that is a compact subset of . Proof: Let be an open cover of , for some indexing set . If is a finite cover then we’re done, so suppose that … Continue reading

Posted in Analysis, Compact, Sequence | Tagged , , | 4 Comments

September 2001 #4

Problem Statement: Assume is a bounded Riemann Integrable function on . Set . Prove that is uniformly continuous on . Proof: Let such that for every . Let and let . Then for it follows that . We are given that is … Continue reading

Posted in Analysis, Riemann Integrable, Uniform Continuity | Tagged , , | 2 Comments

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

Posted in Analysis, Cauchy, Sequence, Series | Tagged , , | 3 Comments

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

Posted in Analysis, Cauchy, Continuity, Differentiable, Math, MVT, Sequence | Tagged , , | Leave a comment

Winter 2004 #4

Problem Statement: Suppose that is differentiable on the bounded interval . Suppose further that there is a constant such that for every . Show that exists and is finite. Note: When I read this question the first time I thought the … Continue reading

Posted in Analysis, Cauchy, Continuity, Differentiable, Math, MVT, Sequence, Uniform Continuity | Tagged , , | 4 Comments