Real analysis Definition and 524 Threads

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) -...

Homework Statement Suppose that \sumanxn has finite radius of convergence R and that an >= 0 for all n. Show that if the series converges at R, then it also converges at -R. Homework Equations The Attempt at a Solution Since the series converges at R, then I know that \sumanRn = M...

I want to start analysis this summer, because I think it'd give me a heads up on upper year quantum mechanics and differential geometry. About me? I'm entering 3rd year physics, and have done intermediate level calculus 1,2,3+vector analysis, linear algebra 1&2 (up to Jordan canonical), and...

Homework Statement Show that limsup(s_n + t_n) <= limsup(s_n) + limsup(t_n) for bounded sequences (s_n) and (t_n). Homework Equations The Attempt at a Solution My book gives a hint that says to first show that sup{s_n + t_n : n > N} <= sup{s_n : n > N} + sup{t_n : n > N}. I'm not...

I would just like to know which of these math courses is best suited for physics. I have taken advanced calculus and linear algebra, so I've seen most of the proofs one typically sees in an intro analysis course (ie. epsilon delta etc.). I intend to do work with a lot of Quantum Field Theory...

Homework Statement Let (s_n) and (t_n) be bounded sequences of nonnegative numbers. Prove that limsup(s_n*t_n) <= (limsup(s_n))*(limsup(t_n)). Homework Equations The Attempt at a Solution I know that (s_n) and (t_n) have convergent subsequences, but that's about it. Where could I go...

Homework Statement Let f(x) = 1 for rational numbers x and f(x) = 0 for irrational numbers. Show that f is discontinuous at every x in R. Homework Equations Definition of continuity. The Attempt at a Solution I want to find a sequence (x_n) that converges to x_0 but that x_n is...

Homework Statement Given that b_{n}\rightarrow\infty and \frac{a_{n}}{b_{n}}\rightarrow C (where C>0) as n\rightarrow\infty, prove that a_{n} must also diverge to \infty, that is, a_{n}\rightarrow\infty as n\rightarrow\inftyHomework Equations As above. The Attempt at a Solution I could...

Homework Statement Using a delta epsilon method prove: \mathop {\lim }\limits_{x \to 1 } x^3+2x^2-3x+4= 4 The Attempt at a Solution I got so far as breaking the equation into =|x||x+3||x-1| now how do I bound it? Also, even more basic question, once I found the bound how do I put the...

Homework Statement f(x) = 4 for x > or = 0, f(x) = 0 for x < 0, and g(x) = x^2 for all x. Thus dom(f) = dom(g) = R. Homework Equations a. Determine the following functions: f+g, fg, f o g, g o f. Be sure to specify their domain. b. Which of the functions f, g, f+g, fg, f o g...

Two problems, actually, but they are very similar. Here goes: Homework Statement Let f be a continuous real-valued function with domain (a, b). Show that if f(r) = 0 for each rational number r in (a, b,), then f(x) = 0 for all x in (a, b). Homework Equations The Attempt at a Solution...

[b]1. Let r(n) = (1+1/n)^n and t(n) = (1+1/n)^n+1. (Use r(n) converge to e). Show that t(n) > r(n) for all n and that lim n->inf(t(n) - r(n)) = 0. Show that {tn} is a decreasing sequence with limit e. {Hint: express {(1+1/n-1)/(1+1/n)}^n as (1+a)^n and apply Bernoulli's inequality). Use n=10...

I found a study page which lists the absolute value of x for x<0 as -x. I think this has to be a typo. The study area is real analysis. Does anyone have better information? Maybe it is some special notation?

Hi, I am running short on time, and I am having problems solving the following questions. I do not have any progress on them yet, but I am currently working on it. So any input is very welcome, even if it is just a basis advice on which direction to go with this stuff. Thanks for your time...

My professor has posted a sample midterm on her web site, but although she promised to post the solutions as well, she hasn't yet and I don't really expect her to at this point since the midterm is tomorrow. I have a few questions about some of the problems on the midterm. The sample exam can be...

Homework Statement Let {a_n} be a monotonically decreasing sequence of positive real numbers with lim a_n = 0. Show the radius of convergence of \suma_nx^{}n is at least 1. The Attempt at a Solution I have no real attempt at a solution since I'm unsure how to proceed. I've tried using...

I am trying to decide which textbook to use to self-study real analysis. I am debating between https://www.amazon.com/dp/0716721058/?tag=pfamazon01-20 and https://www.amazon.com/dp/007054235X/?tag=pfamazon01-20 It seems like Rudin is pretty ubiquitous on the course websites I have looked at...

Give an example of a function f for which \exists s \epsilon R P(s) ^ Q(s) ^ U(s) P(s) is \forall x \epsilon R f(x) >= s Q(s) is \forall t \epsilon R ( P(t) => s >= t ) U(s) is \exists y\epsilon R s.t. \forall x\epsilon R (f(x) = s => x = y) So this was actually a two part question, and...

okay, i already take an intro to proofs class. did average in it. I'm taken real analysis in the spring, but as a degree audit course. the semester following my spring semester , I am take real analysis for a grade.whats the best way to prepare for real analysis?

I'm a math major. I'm looking for the best real analysis textbook that clearly breaks every proof down ,step by step, explaining the purpose of each step , and why you this step is important for the proof.I want a real analysis textbooks that's the subject to comprehend better for all math...

[SOLVED] Real analysis - show convex functions are left &amp; right differentiable Homework Statement Let f:R-->R be convex. Show f admits in every point a left derivative and a right derivative. Homework Equations A function f:R-->R is convex if x1 < x < x2 implies f(x)\leq...

Homework Statement suppose f is a function with the property f(x+y)=f(x)+f(y) for x,y in th reals. suppose f is continuous at 0. show f is continuous everywhere. Homework Equations The Attempt at a Solution I do not understand how to show that f is continuous everywhere.

Homework Statement Suppose that the function f is continuous on [a,b] and X1 and X2 are in [a,b]. Let K1 and K2 be positive real numbers. Prove that there exist c between X1 and X2 for which f(c) = (K1f(X1) + K2f(X2))/(K1+k2) Homework Equations The Attempt at a Solution I...

I have just started in a Real Analysis textbook. It starts "In this chapter we construct the real numbers. We assume that the rational numbers and their arithemtic and order properties are known." What exactly does this assumption mean? Here is an example of where I get caught up. One...

prove \ A_n = ( a, a,a,a,a,...) converges to zero. a \in \ R^p Been reading this real analysis book before i take it next semester and been a lil stuck on this question. I am probably making it seem more difficult than it is. Most of the questions had examples in the chapter but this one...

For my homework, I have to find a counterexample for this: (with S being a subset of the reals.) If P is the set of all isolated points of S, then P is a closed set. I don't quite understand the concept of isolated points, which might be why I can't figure out a counterexample.

So I just wanted to hear about other people's experiences with undergraduate (and introductory graduate) analysis textbooks. There are the standards and some new great texts as well. Which are your favorite? Recommendations? Kenneth Ross: The theory of Calculus, Elementary Analysis. Very...

Homework Statement Use the Intermediate Value Theorem to show that the equation 2^x=3x has a solution c element of (0,1) Homework Equations The Attempt at a Solution Ok I know this theorem is usually very easy, but I've never done one where I couldn't easily solve for x and plug in...

Homework Statement I am having somewhat a difficult time just understanding a simple concept. I am trying to prove that every open subset G of a separable metric space X is the union of a sub collection {Vi} such that for all x belongs to G, x belongs to some Vi (subset of G). I am...

I Apologize in advance for the amount of questions i plan on asking you guys this year. Real Analysis is my first upper division math class and i have not trained my mind to think abstractly enough yet. Homework Statement i) Show that f(x) = x^3 is continuous on R by using...

Homework Statement Prove that the sequence (x_n) = ((a^n+b^n)^{1/n}) converges to b, for 0 < a < b. The Attempt at a Solution I haven't dealt with any sequences with n's in the exponent, but I assume I'll have to use logarithms at some point to get at them? Can someone start me off in the...

Homework Statement The problems states let S1 = 2 and S(n+1) = (square root(2Sn+1)) show that Sn is convergent. Homework Equations The Attempt at a Solution I am not really sure how to go about this proof. Sometimes it is easier to prove something is cauchy than to show it is...

Homework Statement Prove: Let (Xn) be a sequence in R (reals). Then (Xn) has a monotone subsequence. Homework Equations Def: Monotone: A sequence is monotone if it increases or decreases. The Attempt at a Solution I know it has something to do with peak points...that is there...

Let f: X--> Y where X and Y are arbitrary sets. Show the following are equivalent. a. F is 1-1 on X b. f(A/B)=f(A)/f(B) for all subsets A and B of X c. f^-1 f(E) = E for all E that is a subset of X d. f(A intersect B) = f(A) union f(B) for all A,B that is a subset of X I know that in...

I'm having a little difficulty understanding Epsilon in the definition of convergence. From what the book says it is any small real number greater than zero (as small as you can imagine?). Also, since I don't quite grasp what this epsilon is and how it helps define convergence, I am having...

I'm having a little difficulty understanding Epsilon in the definition of convergence. From what the book says it is any small real number greater than zero (as small as you can imagine?). Also, since I don't quite grasp what this epsilon is and how it helps define convergence, I am having...

i'm taking the course of real analysis this semester. However, I find difficulty in understanding it. there're so many terminology and terms. the textbook I'm using is "principles of mathematical analysis" . Sometimes, I read and read to understand a definition or theorem, but still fails to...