Real analysis Definition and 509 Threads

Hi, I have four similar problems that I am not sure how to do: Given: A1 and A2 are in X, B1 and B2 are in Y f: X->Y, g - inverse of f I have to either prove or if false find counterargument 1. f(A1 U A2) = f(A1) U f(A2) 2. f(A1 n A2) = f(A1) n f(A2) 3. g(-1)(B1 U B2) = g(B1) U g(B2) 4...

Hey guys I signed up for a "set theory and topology" class for the fall and was planning on taking real analysis in the spring. Set theory got canceled and so i am taking real analysis instead and pushing set theory to the spring. Is this a wise idea? Taking real analysis before a formal set...

Hi, I am currently a sophomore and a math major with thinking of adding computer science as either minor or second major. I get to register for my classes for Fall Quarter in a week, and I am thinking of taking 2 math classes: One will be numerical analysis, and the other is not yet...

Homework Statement HI all. In order to perform a Fourier transform on a function f(x), f(x) must be integrable, i.e. \int_{-\infty}^{\infty}|f(x)|dx < \infty. Can you confirm that this also implies that f(x) -> 0 for x -> (+/-) infinity?

Homework Statement f(x)=f'(x) for all x in R S.T there exists a c in R such that f(x) = c exp(x) for all x Homework Equations The Attempt at a Solution By defining g = f/c, I was able to show that c= f(0) But i am also supposed to show that c Not equal to any other value I...

Homework Statement Let f: [0,1] -> R be a bounded function. Define S(f,x) = sup { |f(r) - f(s) | : r,s in [0,1] and |r-s| <= x}. Prove that if x>0, y>0 then: S(f,x+y) <= S(f,x) + S(f,y). The Attempt at a Solution I have no idea how to proceed, could you please help?

Homework Statement If Xn is bounded by 2, and |X_{n+2} - X_{n+1}| \leq \frac{|X^2_{n+1} - X^2_n|}{8} , prove that Xn is a convergent sequence. Homework Equations The Attempt at a Solution I believe the solution lies in proving Xn a Cauchy sequence, but I'm not sure how to work it...

Homework Statement Assume S contained in R2 is bounded. Prove that if S is Riemann measurable, then so are its interior and closure 2. The attempt at a solution Proof: If S is Riemann measurable, its boundary is a zero set. Since the boundary of each open U in the int(S) is part of...

Ok this is a proof I'm having trouble with. I've given the solution for the first part I got but I'm stuck and I'm not sure if the first part it correct. any help is appreciated Homework Statement The purpose of this project is to ascertain under what conditions an additive function has a...

Im currently taking an introduction to real analysis class and here's the problem. I do very well with all the math I've encountered before this. I'm really not doing well with the abstract nature of real analysis. I'm having trouble proving things in general because of the fact that i have...

Here is my problem that seems to be easy but I cannot seem to manage to find the answer Homework Statement let f(x) = 1 if x=1/n for some positive integer n = 0 if else the question asks to show f Riemann integrable and find the integral of this function over [0,1]...

Define the sequence A where Asub1=a and Asubn=sqrt(a+Asub(n-1)) for n greater than or equal to 2 I need to determine what positive choices of a will make the sequence converge and to what limit. I also need to prove it. I plugged some values into the sequence and found that it seems to...

Im just starting real analysis and trying to get my mind to start thinking the right way to do these kind of proofs. Any hints would be appreciated. Homework Statement Prove: If |X| is infinite, then |N| <= |X| where N = natural numbers The Attempt at a SolutionI figure you can start by...

im in a very strange situation. i have recently started self-learning real analysis, using rudin and pugh simulatenously, and i am enjoying it a lot. the problems are difficult but i find i can answer most of them (well more than half anyway) yet, when trying to re-learn calculus via spivak...

Learning Real Analysis --> Calculus Hello, Just a quick question I am wondering about. I am going to take my first real analysis course next semester using Rudin. Obviously I have already gone through the usual calculus sequence. I am wondering if learning real analysis will help...

Homework Statement Suppose A is a real nxn matrix and f: R^n --> R is definted by f(v)=v^tAv (where v^t denotes the transpose of v). Prove that the derivative of f satisfies (f'(v))(w) = v^t (A+A^t)w Homework Equations The Attempt at a Solution I'm kinda lost here and I really...

Homework Statement Let y be a fixed real number satisfying 0<y\leq1. Prove that (1+x)^{y}\leq1+ x^{y} for all x\geq0. Homework Equations I'm not sure. The Attempt at a Solution The hint given with the problem states that the derivative of x^{y} is yx^{y-1}. My first thought is...

First, apologies for the notation.. the computer I am currently using does not support the various buttons to create proper notation. Homework Statement Given h(x)=|x| over [-1,1], extend the definition such that h(x+2)=h(x). That is, make h(x) be a periodic sawtooth function. Now let...

Homework Statement 1. Suppose that L1 = lim(x->a+) f1(x) and L2 = lim(x->a+) f2(x) Also, suppose that f1(x)<=f2(x) for all x in (a,b). Show that L1<=L2 2. Suppose f(x) = (sqrt(1+3*x^2) - 1)/(x^2) show that the lim (x->0) f(x) exits and give its value. Homework Equations The...

I have been taking real analysis II this semester and I am starting to get a better grasp over the broad subject of analysis and integration. However, I feel like my understanding is completely problem-oriented. I tried talking to a colleague of mine about real analysis in a conceptual...

Homework Statement f:R->R f satisfies that for all x,y f(x+y) = f(x) + f(y) show that f is continuous on R The Attempt at a Solution I assumed that limit of f at 0 existed. then i showed that that limit must be zero and that f(0)=0, so f is continuous at 0. From there, i broke it up...

I'm not sure how to study for the test since the midterms are open book and open notes. I've been going over the homework, but are there other ways of studying? I'm going over covering compactness, differentiation and integration for the next midterm. Any addition help/advice/problems to look...

Problem Statements a. if a sequence is montone then every one of its subsequences is monotone. true. b. if a sequence is not monotone, then every one of its subsequences is not monotone false. c. if a sequence is unbounded, then every one of its susequences is unbounded. true...

Homework Statement Prove that f(x) = 3*x + 11 is uniformly continuous. Homework Equations x,y in S and |x-y| < a imply |f(x)-f(y)| < e The Attempt at a Solution following the book I get to f(x) - f(y) = 3(x-y) i just don't know how to chose e in this case.. would it...

Homework Statement 1. f is a continuous real valued function on a closed interval. f assumes minimum fvalues on [a,b], there exit x_o,y_o in [a,b] such that f(x_o) <= f(x) <=f(y_o) for all x in [a,b]. Show that the function -f assumes its maximum at x_o in [a,b], then f assumes its minimum...

Homework Statement Suppose f:(a,b) to R(Real numbers), is differentiable at any point x that is in the interval (a,b). Prove that lim(as h goes to 0) of (f(x+h)- f(x-h))/(2h) exists and equals the derivative of f(x) ( which is f '(x)). Give an example of a function where the limit exists...

Homework Statement Show that for any L\in[-1,1] there exists a subsequence of cos(n) such that that subsequence converges to L The Attempt at a Solution I have no idea. I suppose that the ultimate goal would be to find a subsequence nk so that nk converges to x, where x = cos-1(L) + 2\pik

Homework Statement Suppose f is continuous on [0,2]and thatn f(0) = f(2). Prove that there exists x,y in [0,2] such that |y-x| = 1 and f(x) = f(y) Homework Equations The Attempt at a Solution I got the following 1 line proof. Suppose g(x) = f(x + 2) - f(x) on I = [0,2]...

Homework Statement Show that there exists measurable functions f_n defined on some measure subspace, st f_n-> f a.e. but such that f is not measurable. Homework Equations Converges a.e. means that converges everywhere except on a set of measure zero. The Attempt at a Solution Need...

Homework Statement Prove that the interval [0,1] is not a zero set. 2. The attempt at a solution Assume for contradiction that the interval [0,1] = Z is a zero set. This mean that given epsilon greater than 0, there is a countable coverage of Z by open intervals (ai, bi) (___I don't...

Homework Statement Let I be an interval in the real line, and let f map I --> R be a one-to-one, continuous function. Then prove that f^(-1) maps f(I) --> R is also continuous The Attempt at a Solution I've started with the definition of continuity but I don't see where to go next.

Homework Statement Let f be of class C1 on [a, b], with f(a) = f(b) = 0. Show that \int_a^b xf(x)f'(x)dx = -1/2 \int_a^b [f(x)]^2 dx.Homework Equations If F is an antiderivative of f, then \int_a^b f(t)dt = F(b) - F(a) The Attempt at a Solution I'm just really not sure how to begin this one. I...

Homework Statement Let X = (xn) be a sequence of positive real numbers such that lim(xn+1 / xn) = L > 1. Show that X is not a bounded sqeuence and hence is not convergent. Homework Equations Definition of convergence states that for every epsilon > 0 there exist some natural...

Homework Statement Are Lebesgue measures considered metrics? The Attempt at a Solution I have an elementary understanding of metrics and am currently learning about Lebesgue Integration. A Lebesgue measure as a length in Euclidean space should be a metric, but what about areas and...

Homework Statement Find the limit, as n -> infinity, of \sum_{k=1}^nk3/n4 Homework Equations Riemann sum: S(f, \pi, \sigma) = \sum_{k=1}^nf(\xi)(xk - xk-1) The Attempt at a Solution My guess is that I should try to put this sum in terms of a Riemann sum, and then taking n -> infinity will...

Homework Statement Let E be nonempty subset of R which is bounded above (thus, a = sup E exists) Does there exist a strictly monotone sequence in E which converges to a? Homework Equations The Attempt at a Solution I've been thinking about just taking a monotone bounded (this...

what is real analysis? is it only the proofs of calculus or something else? also what are the prerequisites? i have calculus I-II and linear algebra. the course is called intro to analysis and is a year long. the description says it will cover all of rudins principles of math. analysis...

Homework Statement Show that the series \sum(-1)n(x2+n)/n2 is uniformly convergent on every bounded interval in R, but is not absolutely convergent for any x. Homework Equations Weierstrass M-Test The Attempt at a Solution Take g_n(x) = (-1)n(x2+n)/n2. Then |g_n(x)| = (x2+n)/n2. To be...

I am really having a hard time in this intro to real analysis class. I feel as if I'm the only one in class who isn't getting it. I have an extremely hard time thinking abstractly and constructing my own proofs. I know I need a lot of practice. Here is the problem we have to prove:Claim: Let A...

1. Let a be a positive number. Prove that for each positive real number x there is an integer n such that na≤x≤(n+1)a. I have been looking through mounds of books, but haven't figured out where to start. Our teacher just left us hanging on how to figure it out. I am severely stuck and need help...

This is the only way I knew how to get all the math symbols in one document. I hope the link works. http://www.4shared.com/file/63754067/34c48bd5/SupremumInfimum.html" I've done all my work there on the document. Any help, corrections, tips, suggestions are greatly appreciated. Thank...

Which course is more difficulty in terms of which subject contains more rigorous proofs, Complex variables or Real analysis. I don't know whether I should dropped Complex variables, but the only reason I am taken it is because of the useful physics applications found in this course. I my...

Hi all, I'm starting my second year at my uni pretty soon, and I'm trying to make a final decision about the classes that I'm going to take. I still haven't figured out what I want to do in future yet, but I'm thinking of along either academics (not necessarily math, but math is one of my...

Which one should I take first? Does it help to take one before the other?

Homework Statement where A and B are sets and ' \ ' means set difference under what conditions does A \ (A \ B) = B Homework Equations S\A = {X | X in S and X not in A} B < A (subset) means if x is in B then x is in A The Attempt at a Solution I get that if B is a subset of...

Hi, my university is offering a honors version of real analysis based off of Rudin's principals of mathematical analysis. I've heard that Spivak is good preparation so I bought it. It has 30 something chapters so I don't know if I can go through it all. Generally, for anybody who knows, how much...

Hi everyone. I'm taking my first course in real analysis soon. I have a bit of linear algebra and advanced calculus under my belt. I would like to get a book as a companion to the course. What would you recommend? I've heard wonderful things about Rudin's Mathematical Analysis, but is it...

Solved: Real Analysis, differentiation Homework Statement If g is differentiable and g(x+y)=g(x)(g(y) find g(0) and show g'(x)=g'(0)g(x) The Attempt at a Solution I solved g(0)=1 and I got as far as g'(x)=\lim_{\substack{x\rightarrow 0}}g(x) \frac{g(h)-1}{h} but now I...

Homework Statement Suppose a \in \mathbb{R}, f is a twice-differentiable real function on (a, \infinty) and M_0,M_1,M_2 are the least upper bound of |f(x)|,|f'(x)|,|f''(x)|, respectively on (a,\infinity). Prove that M_1^2\leq 4 M_0 M_2Homework Equations The Attempt at a Solution That is...

Homework Statement Prove that if fn -> f uniformly on a set S, and if gn -> g uniformly on S, then fn + gn -> f + g uniformly on S. Homework Equations The Attempt at a Solution fn -> f uniformly means that |fn(x) - f(x)| < \epsilon/2 for n > N_1. gn -> g uniformly means that |gn(x) -...