Real analysis Definition and 509 Threads

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?

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...

Coming up with counterexamples is hard. So to prove or not to prove, that depends if there exists a counterexample. Question 1 has been ANSWERED!: If f has a bounded variation on [a,b] , then is it true that f is of Riemann integration on [a,b]? Question 2 has been ANSWERED!: Is it...

Hello! I am thirteen and self studying real analysis... I'm wondering where to go after this; kinda looking for a general outline of what to do from here on out. Any help is greatly appreciated. Thank you.

So I have an exam in Real Analysis I coming up next week and I was hoping if someone can help me out. I hope my question makes sense because I think I might be confused with defining the metric space or so... Homework Statement a)Suppose that we have a metric space M with the...

Can anyone give me any help on how to get started, or how to do this problem? --- Prove that if the terms of a sequence decrease monotonically (a_1)>= (a_2)>= ... and converge to 0 then the series [sum](a_k) converges iff the associated dyadic series (a_1)+2(a_2)+4(a_4)+8(a_8)+... =...

I know that AP Calculus BC is very watered down in comparison to a real analysis course at the college level. I'm taking high school Calc I, II this upcoming school year as a junior and would like to get a good book to learn real analysis from. I'm currently looking at getting Spivak's CALCULUS...

I'm planning to learn real analysis in the up and coming holidays, anybody have any good suggestions on which textbooks will be useful? I've heard good comments about Baby Rudin, is this true?

Okay, I am a junior undergraduate in CS & Mathematics and I want to be prepared for these two classes when I take them. I am currently signed up in the fall for Real Analysis I and Modern Algebra I (among other classes). I have a strong background in Calculus, but don't have much of a...

Please help me with this one: Suppose f is a continuous real valued function on R - real #s and that 0<f(x)<3 for every x in R. Let F(x) = integral from 0 to x of f(t) dt. a) Show that F is uniformly continuous on R b) Suppose the continuity assumption is left out, but the function f is...

Consider the series 1+ Σ((1/(2^k))coskx + (1/(2^k))sinkx) (a) Show that series converges for each x in R. (b) Call the sum of the series f(x) and show that f is continuous on R = real numbers My thoughts: From trig => cos + sin = 1. So, is it something like |coskx + sinkx| / |2^k| < or =...

Suppose f: R->R is continuous on all of R and B is bounded subset of R. a) show cl(B) is bounded set b) show image set f(B) must be bounded subset of R c) suppose g:B->R is defined & continuous on B but not necessarily on all of R - real #s, Must g(B) be bounded subset of R? (Prove or give...

Hello friends, I wish to seek your assistance in helping me solve these problems regarding Real Analysis from Rudin's Real and Complex Analysis book. All problems are from the first chapter on Abstract Integration. While, I don't expect complete solutions, I hope you guys could give me some...

I'm doing this problem in the book - their are 2 of this kind and they have no answers in the back.. so i thought ill post one. Let S be the Cartesian coodinate plane R x R and define a relation R on S by (a,b)R(c,d) iff a+d=b+c. Verify that R is an equivalence relation and describe the...

Let A and B be subsets if a universal set U. Prove the following. a) A\B = (U\B)\(U\A) To do this, show it both ways. 1) A\B contains (U\B)\(U\A) 2) (U\B)\(U\A) contains A\B I'll start with 2) if x is in (U\B)\(U\A), then x is in (U\B) and x is NOT in (U\A). then (x is in U and x...

Prove: For every real number x>1, there exists two distinct positive real numbers y and z such that x = (y^2 +9)/(6y) = (z^2 +9)/(6z) Okay.. this has real got me beat. Firstly (this sounds stupid and obvious), when they give us a proof, is it really true? Do we just naturally believe...

Hey, university started and in less than a week, we allready have to hand in an assignemnt. I did most questions, i was just wondering if somebody can help check some of work - its more english than math :mad: Q1) True or False - A statement and its negation may both be false. A1) False -...