1. Let (Jn) be a nested sequence of open intervals: J2 J2 2 J3 2.... Is...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
∞ 1. Let (Jn) be a nested sequence of open intervals: J₁2 J2 2 J3 2.... Is J, always nonempty? If your answer is yes, give a proof; if your answer is no, give a counterexample. (Hint: consider an exercise from the last homework.) n=1 2. Let (Jn) be a strictly shrinking nested sequence of open intervals. i.e. Jn = (an, bn), an <bn and an <an+1, bn > bn+1 for all n € N. Prove that Jn #o. (Hint: connect to the nested interval property for closed intervals.) 3. Prove the following limits by definition. (a) lim n non +3 1 n=s (1.1)" 1 (b) lim (c) lim x ². (d) lim -n n = = 0. 1. = 0. 818 n²+1 1. n=1 4. Let C> 0 be a constant. Prove that lim = L if and only if the following statement is true: 87x For every > 0, there exists NEN such that whenever n > N, it is true that xn-L<Ce. 5. Prove that lim n = L if and only if the following statement is true: 1148 For every > 0, there exists NEN such that whenever n > N, it is true that xn-L≤e. ∞ 1. Let (Jn) be a nested sequence of open intervals: J₁2 J2 2 J3 2.... Is J, always nonempty? If your answer is yes, give a proof; if your answer is no, give a counterexample. (Hint: consider an exercise from the last homework.) n=1 2. Let (Jn) be a strictly shrinking nested sequence of open intervals. i.e. Jn = (an, bn), an <bn and an <an+1, bn > bn+1 for all n € N. Prove that Jn #o. (Hint: connect to the nested interval property for closed intervals.) 3. Prove the following limits by definition. (a) lim n non +3 1 n=s (1.1)" 1 (b) lim (c) lim x ². (d) lim -n n = = 0. 1. = 0. 818 n²+1 1. n=1 4. Let C> 0 be a constant. Prove that lim = L if and only if the following statement is true: 87x For every > 0, there exists NEN such that whenever n > N, it is true that xn-L<Ce. 5. Prove that lim n = L if and only if the following statement is true: 1148 For every > 0, there exists NEN such that whenever n > N, it is true that xn-L≤e.
Expert Answer:
Answer rating: 100% (QA)
No a counterexample is Jn 1n 2n for n 1 Let In an bn be ... View the full answer
Related Book For
Introduction to Real Analysis
ISBN: 978-0471433316
4th edition
Authors: Robert G. Bartle, Donald R. Sherbert
Posted Date:
Students also viewed these mathematics questions
-
Let (Sn) be a nested sequence of nonempty closed subsets of a complete metric space with d(Sn) 0. Their intersection contains exactly one point.
-
Let (In) be a nested sequence of closed bounded intervals. For each n N, let xn In. Use the Bolzano-Weierstrass Theorem to give a proof of the Nested Intervals Property 2.5.2.
-
Let S1 S2 S3... be a nested sequence of nonempty compact subsets of a metric space X. Then S-542 i=1
-
1. Using a diagram of utility maximization, show: a. The income effect of price change b. The substitution effect of price change
-
An automatic valve consists of a 225 Ã 225-mm square plate that is pivoted about a horizontal axis through A. If the valve is to open when the depth of water is d = 450 mm, determine the...
-
The tank is accelerated upward at a uniform rate. Does the manometer level go up or down? P = 2 psig
-
The L. Ming Company has had great difficulty controlling costs in Singapore during the past three years. Last month a standard cost and flexible-budget system was installed. Results for a department...
-
Buddy Holly Companys sales budget projects unit sales of part 198Z of 10,000 units in January, 12,000 units in February, and 13,000 units in March. Each unit of part 198Z requires 2 pounds of...
-
Tutorial Exercise Evaluate the integral using the following values. L x3 dx = 1484, Lx dx dx = 28, dx = 4 dx 2 Step 1 Now use the following to evaluate the integral L(-1) +1 dx. By basic integration...
-
Peru Furniture City, Inc., sold $88,000 of furniture in May to customers who used their American Express Credit Cards. Such sales are subject to a 5% discount by American Express (a nonbank credit...
-
Explain the relationship of self-leadership to leadership competency. How does a leader's self-leadership skills impact those they lead? Identify one self-leadership skill that you consider to be a...
-
Exercise 10-5A (Algo) Journal entries for a line of credit LO 10-2 Singer Company has a line of credit with United Bank. Singer can borrow up to $440,500 at any time over the course of the Year 1...
-
Why did vaccine companies collaborate with governments in the U.S. and the UK during the global COVID-19 pandemic, as noted in the case?
-
why did healthcare historians such as nightingale place dvidenced data on thd back burner?
-
K 22 Find the length of the parabolic segment r= 1+ cos 0' 2' The length is (Type an exact answer, using radicals as needed.)
-
Why do critics suggest that IQ tests are not a reliable predictor of performance?
-
Detailed explanation for answer (excel breakdown if possible). ABC, Inc.'s revenues have been $300,000 and total costs have been $150,000; both costs and revenues are expected to remain the same in...
-
Determine the volume of the parallelepiped of Fig. 3.25 when (a) P = 4i 3j + 2k, Q = 2i 5j + k, and S = 7i + j k, (b) P = 5i j + 6k, Q = 2i + 3j + k, and S = 3i 2j + 4k. P
-
(a) Prove there is no n N such that 0 < n < 1. (Use the Well-Ordering Property of N.) (b) Prove that no natural number can be both even and odd.
-
Establish the convergence or the divergence of the following integrals: (a) (b) (c) (d) (e) (f) In x dx rdx o (r1) dx
-
Let I := [a, b] and let f : I R be continuous on I. If f has an absolute maximum [respectively, minimum] at an interior point c of I, show that f is not injective on I
-
Which of the following is NOT a common type of white box test of controls? a. completeness tests b. redundancy tests c. inference tests d. authenticity tests
-
In an automated payroll processing environment, a department manager substituted the time card for a terminated employee with a time card for a fictitious employee. The fictitious employee had the...
-
Which of the following is NOT a test for identifying application errors? a. reconciling the source code b. reviewing test results c. retesting the program d. testing the authority table
Study smarter with the SolutionInn App