What is Real analysis: Definition and 509 Discussions

In mathematics, real analysis is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.
Real analysis is distinguished from complex analysis, which deals with the study of complex numbers and their functions.

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  1. S

    Should I have learned real analysis before taking complex analysis

    Hi I am taking summer class in complex analysis and I am having a horrible time. I don't understand anything we've covered so far, e.g. Cauchy-Goursat theorem, Laurent series, series expansion, etc. The prereqs was just Calc III, which I got an A- in. The textbook isn't much help...
  2. M

    Real Analysis - Natural Number Induction

    Homework Statement Prove that if n is a natural number greater than 1, then n-1 is also a natural number. (Hint: Prove that the set {n | n = 1 or n in \mathbb{N} and n - 1 in \mathbb{N} } is inductive.)Homework Equations The Attempt at a Solution S(n) = \sum_{j = 2}^{n} j = 2 + 3 + \cdots...
  3. J

    Real Analysis Proof (Limits of Functions)

    Let A be a subset of ℝ. Let c be a limit point of A. Consider the function f: A → ℝ Claim: If the function f has not have a limit at c, then there exists a sequence (xn), where xn≠c for all n, such that lim xn=c, but the sequence (f(xn)) does not converge. Since the function f does not have a...
  4. B

    A good introductory book on real analysis with examples

    Hello I am wonderinf if you guys know about a good introductiry book on real analysis where they use examples. I have found some books online, but they seem to not show with examples what we can do. I have read to Spivak calculus book is a good introduction, but I think it was too basic. I...
  5. PeteyCoco

    Real Analysis after Multivariable Calculus a bad idea?

    I studied from Multivariable Calculus by James Stewart this past year and thought that it would be worth reading another calculus text to fill in the gaps and to keep my skills sharp. While reading Advanced Calculus by David Widder, I came across this problem: (Paraphrased from text) Suppose a...
  6. S

    Real analysis epsilon delta problem

    I've been reading through Spivak's calculus, and the problem is the answer key i have a hold of is for a different edition so it often doesn't answer the correct questions. Anyways, here they are: Chapter 5 problem 10 b. Prove that lim x-> 0 f(x) = lim x-> a f(x-a) c. Prove that lim...
  7. W

    Real Analysis: Understanding Proof with Equations 3 & 4

    Hi all, I am trying to understand this basic proof but I don't understand that where the equations (3) & (4) have come from? [img=http://s9.postimg.org/8nwoy04vj/image.jpg] p.s. sorry if I have posted this thread on wrong website.
  8. B

    So, How can we use algebraic manipulation to prove the limit of a sequence?

    I am trying to prove that \lim\left[\sqrt{n^2+n}-n\right]=\frac{1}{2} Where n \in \mathbb{N} and \lim is the limit of a sequence as n\to\infty. From the definition of a limit, I know that I need to show that \exists{N}:n>N\Rightarrow\left|\sqrt{n^2+n}-n-\frac{1}{2}\right| < \epsilon...
  9. Government$

    Best way self study real analysis?

    In October of this year i will start with math major and i decided to prepare myself in spear time. In my faculty there is no such thing as Calculus but rather you go strait to the analysis and you pick up calculus along. (There is singe variable calculus in high school). In first two years...
  10. STEMucator

    What book to use for a first timer trying to learn real analysis?

    Hey, I recently just finished my calc 2 course ( All my exams actually :) ), and I'm thinking about learning real analysis over the summer. Just to stay keen and it's a topic I'm really interested in. I've heard Rudin is a staple ( heard it was tough as well ), but I would like to hear other...
  11. J

    MHB Bounded linear functional question? Real Analysis

    Consider the functional Tf = f(5) - i f(7). If we take the domain T to be C_0(ℝ) with supremum norm, is T a bounded linear functional? What if we take the domain to be C_c(ℝ) with L^2 norm || . ||_2?I know I should post what I have so far but this time I have no idea because I had to missed 2...
  12. caffeinemachine

    MHB Problem book for real analysis

    Hello MHB, I want to sharpen my analysis skills. I am looking for a problem book in real analysis. Anybody here knows a good one? I use Rudin's Principle of Mathematical Analysis for reading.
  13. phosgene

    Job prospects: real analysis vs statistics

    Hi guys, I'm entering my second year of a science degree majoring in physics. I'm torn between taking real analysis or statistics this year. Which ever one I take will heavily influence my second major (if I don't take real analysis then its statistics or applied maths. If I do take it then it's...
  14. Astronuc

    Analysis Real Analysis: Modern Techniques & Their Applications, Gerald Folland

    Author: Gerald B. Folland Title: Real Analysis: Modern Techniques and Their Applications (Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts) Amazon Link: https://www.amazon.com/dp/0471317160/?tag=pfamazon01-20 Prerequisities: Calculus, linear analysis, complex...
  15. micromass

    Analysis What are the recommended books for learning Real Analysis by Anthony Knapp?

    Author: Anthony Knapp Title: Basic Real Analysis Amazon link https://www.amazon.com/dp/0817632506/?tag=pfamazon01-20 Prerequisities: Calculus, proofs Level: Undergrad
  16. A

    Euler's Method of Trigonometric Series- Real Analysis

    Homework Statement Use Equation (1) to derive the formula: (1-a*cos(phi))/(1-2*a*cos(phi)+a^2))= 1+a*cos(phi)+a^2(cos(2phi))+a^3(cos(3phi))... Homework Equations Equation (1) is 1/(1-a(cos(phi)+isin(phi)))= 1+ [a(cos(phi)+isin(phi)]+ [a(cos(phi)+isin(phi)]^2 +...
  17. C

    Is real analysis helpful in physics?

    I have tons of room for electives and I'm filling them up with math classes. I can see how complex analysis and even abstract algebra would be helpful, but would real analysis be helpful for research in the future? Other than getting a deeper understanding and proof based re-introduction to...
  18. micromass

    Analysis What topics are covered in Real Analysis by Carothers?

    Author: Carothers Title: Real Analysis Amazon link https://www.amazon.com/dp/0521497566/?tag=pfamazon01-20 Prerequisities: Being acquainted with proofs and rigorous mathematics. Rigorous calculus. Level: Grad Table of Contents: Preface Metric Spaces Calculus Review The Real Numbers...
  19. Q

    Learning real analysis without linear algebra?

    Well, I am a second year astrophysics student in the UK. However, I want to go for a PHD in theoretical physics after my graduation. So I believe I have to take more maths modules as much as possible. I have taken mathematical techniques 1 and 2 which cover up to vector calculus, differential...
  20. S

    Am I ready for Real Analysis I?

    Hi - I am a second semester Sophomore, and am wondering what I need to know to succeed in Real Analysis. My background is in linear algebra, differential equations, and calculus. However, I have not had any real exposure to rigorous proofs which I hear what you do in RA. The prerequisite for...
  21. S

    Where can I learn real analysis and proving online?

    Hello, I noticed that the most difficult questions in my maths course are usually real analysis and proving questions like http://i.imgur.com/IBols.png and http://i.imgur.com/KETnq.png. I was going to buy university level textbooks to cover them but my teacher told me not to because I need to...
  22. H

    Real analysis: Problem similar to uniformly integrable

    Homework Statement Assume \mu(X) >0 and that f is a measurable function that maps X into ℝ and satisfies f(x) >0 for all x\inX. Let \alpha be any fixed real number satisfying 0<\alpha<\mu(X) <infinity Prove that inf { \int_{E}f d\mu : E\inM, \mu(E) ≥\alpha} >0. (Hint. First prove...
  23. D

    Courses Suggestions on independent study courses/book choices? Real analysis, fourier, ?

    Hey, I was wondering if you guys could offer any course guidance on independent studies I could try to take my senior year. I have some ideas, but I was wondering whether you guys could give me any recommendations/book recommendations. My background: I initially wanted to go into a more...
  24. S

    How hard is Real Analysis 2 compared to Real Analysis 1?

    Is it a lot harder? I'm taking Real Analysis 1 this semester, and am planning on taking the second part to the course in the Winter. Also, would it be a bad idea to take Real Analysis 2 and Elementary Number Theory in one semester? Thanks
  25. M

    Formal Proof of Uniform Continuity on a Closed Interval

    Homework Statement Prove that if f is uniformly continuous on [a,b] and on [a,c] implies that f is uniformly continuous on [a,c]. Homework Equations The Attempt at a Solution This is my rough idea for a proof, can someone help be say this more formally? Is my thinking even...
  26. N

    Real Analysis Mean Value Theorem Proof

    Homework Statement Let f: R->R be a function which satisfied f(0)=0 and |df/dx|≤ M. Prove that |f(x)|≤ M*|x|. Homework Equations Mean value theorem says that if f is continuous on [a,b] and differentiable on (a,b), then there is a point c such that f'(c)=[f(b)-f(a)]/(b-a). The...
  27. M

    Proving Even Fct Lim x->0 f(x)=L iff Lim x->0+ f(x)=L

    Homework Statement Prove that if f: R->R is an even function, then lim x->0 f(x)=L if and only if lim x->0+ f(x)=L. Homework Equations The Attempt at a Solution So far I have: If f is an even function f(x)=f(-x) for x in domain of f. Then I am trying to apply the limit...
  28. T

    MHB Real Analysis is all about infinity

    My lecturer posted a question asking why ""Real Analysis is all about infinity" Why is this so?
  29. D

    Basic Real Analysis: Are Open Intervals Always Roomy?

    Homework Statement This problem starts with a definition, A set S \subseteq R is said to be roomy if for every x \in S, there is a positive distance y > 0 such that the open interval (x - y, x + y) is also contained in S. Problems based on this definition: a) Let a < b. Prove that the...
  30. G

    Real Analysis limit proof problem.

    Homework Statement Define the function f:ℝ→ℝ by f(x)=0 if x is irrational and f(p/q)=1/q if p,q are integers and q>0 and the fraction is in reduced form. Prove f is continuous at every irrational point. Homework Equations The Attempt at a Solution We must show that lim x->a...
  31. G

    Does the Limit Exist for a Piecewise Function at x=1/2?

    Homework Statement f:ℝ→ℝ is defined as f(x)= 2x if x is rational and f(x)=4-2x if x is irrational. Is it true that lim x→1/2=1? 2. The attempt at a solution Intuitively it seems that as x gets ever closer to 1/2 from either side that the function will oscillate between numbers very...
  32. H

    Need a good real analysis book for undergrad intro course

    Need a good real analysis book for undergrad "intro" course I'm a computational math major (double majoring with MechE) and basically I'm required to take an "intro" (400 level) real analysis sequence with the comp. math department. This course is shaping up to be an incredibly nasty surprise...
  33. H

    Continue on to elementary real analysis or review calculus?

    Im kind of rusty on my calculus II and III and I was wondering if I should review it before I try to self teach myself basic real analysis? I have some experience with basic proofs.
  34. U

    Real Analysis Least Upper Bound Question

    Homework Statement If S1, S2 are nonempty subsets of ℝ that are bounded from above, prove that l.u.b. {x+y : x \in S1, y \in S2 } = l.u.b. S1 + l.u.b. S2 Homework Equations Least Upper Bound Property The Attempt at a Solution Using the least upper bound property, let us suppose...
  35. srfriggen

    Prove 1/n < a < n: Real Analysis Homework

    Homework Statement Prove that if a>0, there exists n in N such that 1/n < a < n. Homework Equations The Attempt at a Solution I am starting with a>0 and trying to manipulate, algebraically, to get n > a > 1/n. From a > 0 I can add 1 to both sides to obtain, a+1 > 1. Then I...
  36. S

    Am I ready to take Real Analysis 1?

    I'm a math major, and recently started taking upper level math classes. So far, the only upper levels I've taken in math are Abstract Algebra and Applied Combinatorics. To be honest, I didn't really work as hard as I should have in Abstract, and feel like my proof writing skills are not all...
  37. G

    Real analysis and accumulation points of a series

    I was reading a textbook on real analysis and came across this definition:Given a real sequence we say x is an accumulation point if given any \in greater than 0 we can find infinitely many natural numbers n such that |xn-x| is less than \in. I also found a theorem that stated if a real...
  38. L

    Anyone here taken Real Analysis yet?

    Real Analysis will be the most rigorous, proof-based course I've taken for my math major, and I'm concerned because a lot of people at my school HATE the course. Any tips on preparation? Surviving?
  39. P

    Taking Topology, Real Analysis and Abstract Algebra concurrently a good idea?

    Hello all, In the Fall I am planning on taking Real Analysis, Abstract Algebra and doing an independent study in something(my professor has yet to get back to me on what he is willing to do it in). My question is would it be too much of a workload to try and do another independent study in...
  40. ?

    400-level versus 500-level Real Analysis

    I'm preparing to start a year-long sequence of 400-level real analysis using Rudin's Principles of Mathematical Analysis 3E in my second undergrad year, and my advisor recommends I take the graduate-level sequence the following year through Real Analysis by Stein and Shakarchi. Since both of...
  41. S

    Numerical Analysis and Real Analysis in one semester?

    Hi, I am thinking of taking Intro to Numerical Analysis and Real Analysis 1 course next semester, but was wondering if maybe that'll be too much of a load? Is numerical analysis a tough course? These courses will be taken alongside 2 other statistics courses and maybe a history class...
  42. M

    A Real Analysis question on anti-derivatives

    Let f : R to R be a continuous function, and assume anti-derivative of f(x)dx from m to n≤ (n-m)^2 for every closed bounded interval [m,n] in R. Prove that f(x) = 0 for all x in R. I tried using fundamental theorem of calculus but got stuck. Any help/suggestion would be appreciated.
  43. J

    Evaluate this limit, introductory real analysis

    Homework Statement limit of the sequence, [xn]=(-3n2+n+1)/(n2-2n+3) Homework Equations I so far know about the definition of a limit, squeeze principle, and lim[xnyn] = 0 if xn or yn goes to 0 The Attempt at a Solution Tried the definition of the limit but the algebra got really...
  44. G

    Real Analysis - Differentiation in R^n - Example of a specific function

    Homework Statement Give an example of a continuous function f:R^2→R having partial derivatives at (0,0) with f_1 (0,0)≠0,f_2 (0,0)≠0 But the vector (f_1 (0,0),f_2 (0,0)) does not point in the direction of maximal change, even though there is such a direction. (If this is too difficult...
  45. A

    Have I proved this obvious fact correctly? (Real Analysis)

    Homework Statement It's not a HW problem. I was reading baby Rudin, in chapter 6 when it talks about Riemann–Stieltjes integral, it claims that given ε>0, we could choose η>0 such that (α(b)-α(a))η<ε. I wonder why it is true. I proposed this question to myself: Suppose that ε>0 is an...
  46. T

    Real Analysis: proving a sequence converges and finding its limit.

    Homework Statement Suppose r>1. Prove the sequence \sqrt[n]{1 + r^{n}} converges and find its limit. Homework Equations The Attempt at a Solution It's obvious that the sequence converges to r, so I know where I need to end up. My first instinct is to use the squeeze theorem...
  47. R

    Real analysis, simpl(er) questions

    Homework Statement It is a 4 parter, but i got 3 and 4 done. a) Find f ([0,3]) for the following function: f(x)=1/3 x^3 − x + 1 b) Consider the following function : f(x) = e^(−ax) (e raised to the power of '-a' times 'x') a, x ∈ [0,∞) Find values of a for which f is a contraction ...
  48. R

    Intro To real analysis problem

    Homework Statement a) Find f ([0,3]) for the following function: f(x)=1/3 x^3 − x + 1 b) Consider the following function : f(x) = e^(−ax) (e raised to the power of '-a' times 'x') a, x ∈ [0,∞) Find values of a for which f is a contraction . c) Prove that for all x,y ≤ 0 | 2^x −2^y | ≤ |x−y|
  49. R

    Real Analysis, Sequence/series/supremum/infimum

    Homework Statement a) Given the definition of the divergence of a sequence {a_n} : "For any H >0 we can find a number NH such that a_n >H, for all n>N_H" prove that {a_n * b} diverges if {a_n } diverges for any b ≠ 0 . b) Find the supremum and infimum for the se… 1 - 1/n } and, if...
  50. R

    Convergence and Divergence Tests for Series: Real Analysis Homework

    Homework Statement a) Show that the series ∑ from n = 1 to infinity 1/n^p where p converges when p > 1 and diverges for p=1. b) Prove that the following series diverges: ∑ from n = 1 to infinity sqrt(n)/n+1 c) Use an appropriate test to show whether ∑ from n = 1 to infinity [(−1)^n *...
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