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Real Analysis

Real Analysis: Continuous Functions

Theorem: Let E and E' be metric spaces, with E compact and E' complete. Then the set of all continuous functions from E to E', with the distance between two such functions f and g taken to be max {d'( f(p), g(p) ) : p is an element of E} is a complete metric space. A sequence of points of this metric space converges if and only

Continuous Functions Metric Spaces

Let S be a subset of the metric space E with the property that each point of eS is a cluster point of S (one then calls S dense in E). Let E' be a complete metric space and f:S->E' a uniformly continuous function. Prove that f can be extended to a continuous function from E into E' in one and only one way, and that this extended

Real Analysis: Metric Spaces

Give an example of each of the following: a) an infinite subset of R with no cluster point b) a complete metric space that is bounded but not compact c) a metric space none of whose closed balls is complete

Closure, Interior and Boundary: Example Problem

What are the closure, interior, and boundary of an open ball in E^n? A closed ball in E^n? What are the closure, interior, and boundary of a subset S of E with the discrete distance function (d(p,q)=1 unless p=q)?

open sets in this metric space

Let E be an arbitrary set and, for p,q (elements of) E, define d(p,q) = 0 if p = q, d(p,q) = 1 if p does not equal q. This is a metric space. What are the open and closed balls in this metric space? Show that two balls of different centers and radii may be equal. What are the open sets in this metric space?

Real Analysis - Sets & Proofs

Let S be a subset of the metric space E. A point p (element of ) S is called an interior point of S if there is an open ball in E of center p which is contained in S. Prove that the set of interior points of S is an open subset of E (called the interior of S) that contains all other open subsets of E that are contained in S.

Value of the Limit: Limit of Function

(14) Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places). lim (x->16) (squareroot of x)- 4/(x-16), x = 17, 16.5, 16.1, 16.05, 16.01, 15, 15.5, 15.9, 15.95, 15.99.

Sum of Series Using Taylor Expansion

Begin with Taylor series for the cosine and derive the equation: Sum (1/n^2) = Pi^2/8 Use the previous result and a smart choice of x to get that: sqrt(2) = (2*2*6*6*10*10....)/(1*3*5*7*.....). See the attached file.

Real Analysis Textbook Metric Spaces

Please solve the problems in B.pdf file by using the textbook. The topics covered in this problem set are: 1. Metric spaces. Basic concepts ( Sections 5.1-5.2 ) Problems: - Section 5--# 1,2,4,5,6,7 2. Convergence. Open and closed sets (Sections 6.1-6.6) Problems: - Section 6--# 1,2,3,4,5,9,10 3. Complete m

One-sided Limits and Continuity

Limit Find the indicated one-sided limit. If the limiting value is infinite, indicate whether it is +∞ or -∞. 1. 2. and lim f(x) where f(x)= Decide if the given function is continuous at the specific value of x and why? 1. f(x) = 2. f(x)= Please see the attached file for the fully formatted problem

Series - ratio test

I need help determining how to calculate the limit of a given series using the ratio test (particular problems I am struggling with are attached) Thanks for your help! a)Use the ratio test to decide if the series converges or diverges. Let L be the limit obtained when applying the ratio test. Determine the value of L.

Real Analysis: Topological Spaces

In the textbook: Topological spaces 1.Compactness in Metric spaces (Sections 11.1-11.4) Problems: - Section 11 --# 1,2,3,4 2.Real functions on Metric and Topological Spaces (Sections 12.1.12.2) Problems: - Section 12 --# 1,2,3,5,6

Function Convergence Proofs

** Please see the attached file for the complete problem description ** Let f be a function defined on R and, for each natural number n, define the function f_n by.... Decide whether or not you believe the statement is true.

Divergence and Convergence

** Please see the attached file for the complete problem description ** Please show all work (see attached) 1) Use integral test to determine if a series converges or diverges. Do not calculate what exact value series converges towards if it does converge. 2) Use integral test to determine if series converges or diverge

Limits

A. Let f: R--->R and let c be element in R. Show that the lim from x to c of f(x)=L if and only if lim from x to 0 of f(x+c)=L (if and only if: go both ways) b. Use either the epsilon-delta definition (which states: Let A be a subset of the reals and let c be a cluster point of A. For a function f: A--->R, a real number L is

Real Analysis Functions

My questions are from sections 5 and 6. Since most of questions are from section 6, I will just just attach reference_6 pdf file with my question sheet. I think you actually have all of my reference copies. This is the same on-line link to my textbook that I sent you earlier today: http://books.google.com/books?id=z8IaHg

Lipschitz functions convergence

From Convergence, Open, and Closed Sets I am attaching my reference pdf file. Please prove #11 on page 55 based on my reference. Let M_X be the set of all functions f in C_[a,b] satisfying a Lipschitz condition.....

Metric space real analysis

From Contraction Mapping. The link to my textbook: http://books.google.com/books?id=z8IaHgZ9PwQC&pg=PA72&dq=unique+solution+contraction+mapping&lr=#PPA66,M1 Section 8:Contraction Mapping from page 66 to page 77. Please solve for #5 and #6 based on our reference. If you have any question or suggestion, please contact m

Sequences and limits.

Establish the convergence and find the limits of the following sequences ((1+1/2n)^n)) ((1+1/n^2)^(2(n^2))) ((1+2/n)^n)) Give an example of a convergent sequence Xsubn of positive numbers with lim(Xsubn^1/n)=1 Give an example of a divergent sequence Xsubn os positive numbers with lim(Xsubn^1/n)=1

Complete Metric Spaces

Complete Metric Spaces Problem 1: Prove that the limit f(t) of a uniformly convergent sequence of functions {f_n(t)} continuous on [a,b] is itself a function continuous on [a,b]. Hint. Clearly |f(t) - f(t_0)| < |f(t) - f_n(t)| + |f_n(t) - f_n(t_0)| + |f_n(t_0) - f(t_0)|, where t, t_0 are real numbers of [a,b]. Use t

Convergence and limits problems

Show that the following sequences are not convergent i) (2^n) ii) (((-1)^(n))n^2)) that is negative 1 to the n times n squared Show that if X(sub n) is greater than or equal to zero for all n element N and lim(X(sub n))=0 then the lim(sqrtX(sub n))=0

Set Theory in Real Analysis

Undergraduate senior level Real Analysis. Please show me formal math proofs. Prove that the set of all real functions...defined on a set M is of power greater than the power of M...(see attached)

Real numbers analysis

A. let f(x):= 1/(x^2), x does not equal zero, x elements of Reals determine the direct image f(E) where E= {x elements Reals: 1<=x<=2} determine the inverse image f^-1(G) where G= { x element of Reals: 1<=x<=4} b. show that if a<b then a< 1/2(a+b)<b c. for a,b,c elements of reals with a<b, find an explicit bijection of

Mathematics of Flight Time

Two planes leave O'Hare Airport in Chicago at the same time.One flies directly east at 540 miles per hour.The other flies directly west at 720 miles per hour.How long will it take them to be 1890 miles apart.

Power series solution

Compute the power series solution for the ODE y''+x^2y=0 and determine the radii of convergence for the two series solutions.

Effect of Changing a Company's Credit Terms

The Dunstill company has obtained the following information: ? Annual Credit Sales= 30 million ? Collections period is= 60 days ? Credit terms= Net 30 days ? Interest rate = 12% The company is considering changing its credit terms to 4/10 net 30. The company anticipates 50% of the customers will take advantage of this poli

Convergent Sequences and Limits

Suppose {an} is a convergent sequence and {bn} is a sequence with the property that bn < an if n is even and bn > an if n is odd. Does bn converge? Determine the limit of the given sequence (2 problems). (3n^3+2n+1)/(5n^2-2*n+3) 1/[1+(n^3 / 3^n)] Please see the attached file for the fully formatted problems.