### Real analysis : Maximum-Minimum Theorem

Let I:=[a,b] be a closed bounded interval and let f:I->R be continuous on I. Then f has an absolute maximum and an absolute minimum on I.

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Let I:=[a,b] be a closed bounded interval and let f:I->R be continuous on I. Then f has an absolute maximum and an absolute minimum on I.

Prove OR disprove the following statements. Explain. (i) There is a nonempty set S in R ( real numbers) such that S contains none of its limit-point(s). (ii) There is a nonempty set S in R such that S is neither open nor closed. (iii) There is a nonempty set S in R such that S is both open and closed. (iv) Let a

A). Prove that the function f(x) = e^x is differentiable on R, and that (e^x)' = e^x. ( Hint: Use the definition of e^x, and consider the sequence of partial sums.) My thoughts on a: I tried to prove the differentiability by proving continuity on R, since e^x is series, sum of polynomials, and all polynomials are different

1. Find the 3rd-degree Taylor polynomial for f(x) = 1/(2+x) at a = 1. 2. Comparison Test let . 3. Comparison Limit let . 4. Comparison Test . Please see the attached file for the fully formatted problems.

Prove or disprove the following statement. Provide detailed answer and justify all steps. 1). There is a nonempty set S in R such that closure of S is equal to R and the closure of its complement closure(R-S) also is equal to R. My thoughts on this problem: Q ( rationals) and R-Q (irrationals), but how to prove that the cl

This question is from Advanced Calculus II class, it is more like introduction to real analysis. Let f_n: R -> R be the sequence of functions given by f_n(x) = x/ ( 1 + nx^2) a). Prove that the sequence f_n converges uniformly to a function f. What is f? b). Prove that for each x in R-{0} ( all real numbers but 0 not in

1. Determine whether the sequence {see attachment} converges, and find its limit if it does converge ... 2. Determine whether the given infinite series converges or diverges. If it converges, find its sum ...

Let f be a nonnegative measurable function. Show that (integral f = 0) implies f = 0 a.e. See attached document for notations. Please help: This problem is from Royden's Chap 4 text on Lebesgue Integral.

Show that the sum and product of two simple functions are simple. Show that [Definition of simple: A real-valued function is called simple if it is measurable and assumes only a finite number of values. If is simple and ahs the values then , where .] This problem is from Royden's Real Analysis text for gradu

Let's say you work in a company where over 2000 people are employed. Using the Central Limit Theorem, where the mean age of all employed is 37 with a standard deviation of 13; If 5 people are randomly selected, find the probability of their age being less than 22.

1. For each of the sequences whose nth term is given by the formula below (so of course n takes successively the positive integer values 1,2,3...), does it have a limit as n tends to infinity? In each case, briefly explain your answer including justification for the value of the limit (if it exists) a) (1/3)ⁿ b

1- Find the Taylor series generated by f at X = a. f (x) = 1/(10-x) a = 3 2- Find the Maclaurin series for the given functions. A) 1/(6+x) B) sin 10x.

Find each limit or explain why it does not exist. Please see attachment.

Each series telescopes. In each case - express the nth partial sum Sn in terms of n and determine whether the series converges or diverges. Please see attachment for full question.

Please assist me with the attached problems, including: 8.7 Find the convergence set for the power series ... 8.8 Given the series (a) estimate the sum of the series by taking the sume of the first four terms. How accurate is the estimate? (b) How many terms of the series are necessary to estimate its sume with three-place

Please assist me with the attached problems, including: 1. Sketch each vector 2. Find the standard representation and length of each vector 3. Maclaurin and Taylor Series and Radius of Convergence See attachment for complete list of questions. Thanks.

The definition of average value of a continuous function can be extended to an infinite interval by defining the average value of f on the interval [a, ∞) to be Lim as t approaches ∞ 1/(t-a)integrand from a to t f(x)dx 1. Find the average value of {see attachment} on [0, ∞). 2. Find the lim as x goes t

1. Test for convergence or divergence, absolute or conditional. If the series converges and it is possible to find the sum, then do so {see attachment} 2. Find the open interval of convergence and test the endpoints for absolute and conditional convergence: {see attachment} 3. For the equation f (x) = ... {see attachment

Suppose that a function F(s) has a pole of order m at s=s0, with a Laurent series expansion ... in the punctured disk 0<|s - s0|<R2, and note that (s-s0)ⁿ F(s) is represented in that domain by the power series ... By collecting the terms that make up the coefficient of (s-s0)ⁿֿ¹ in the product (Sec. 61) of

Thank you in advance for your help; this one sounds like it should be simple, but I still continually get the wrong answer: "Evaluate the limit as x goes to infinity of (1+(3/x))^(4x)."

Real Analysis Let and let be continuous map given by .

Real Analysis Let xEn/n be an irrational number. For each rational number yEn define... Let be the collection of all these open balls of n. Is α an

S(1): Let ε=1, and let any δ>0 be given. S(2): Let n be an integer > max(1, 1/δ), and set x=1/n and y=1/(n+1). S(3): Both x and y belong to (0,1), and |x-y| = 1/n(n+1) < 1/n < δ. S(4): However, |f(x)-f(y)| = |n-(n+1)| = 1 = ε S(5): This contradicts the definition of uniform continuity (i.e.,

By an n-fold line subdivision of the plane P, we mean any collection of n-distinct (infinite) lines in P, together with the open regions in P that they determine. (We don't count the lines as part of the regions.) Let us say that two such regions are adjacent if their boundaries have a positive-length or infinite line segment

Real Analysis Divergence Theorem Let V be a region in ⁫3complying with the hypotheses of the divergence theorem, and denote by S its boundary surface. Let also φ: 

1. For i = 1,2 let fi: Xi --> Yi be maps between topological spaces. Show that the product f1Xf2: X1XX2 --> Y1XY2 defined by f1Xf2(x1x2):= (f1(x1), f2(x2)) is continuous if and only if f1 and f2 are continuous. *(Please see attachment for proper representation of formulas and problem #2)

Proving in Real Analysis Course. Please see the attached.

Give formal negations of the following definitions: * Limit point. Your answer should be in the form: "A point p in X is NOT a limit point of the set E in X if ... " * Interior point. Your answer should be in the form: "A point p in X is NOT an interior point of the set E in X if ... " * Closed set. Your answer

Fix a point p in R. Let { Iα } be a ( possibly infinite ) collection of open intervals Iα = ( cα , dα ) which is a subset of R, such that pЄ Iα for all α. Prove that the union I: = Uα Iα is also an open interval ( possibly infinite ). Hint: Cons