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.

View More On Wikipedia.org
Recent contents Top users
  1. i_hate_math

    I Upper bound and supremum problem

    Claim: Let A be a non-empty subset of R+ = {x ∈ R : x > 0} which is bounded above, and let B = {x2 : x ∈ A}. Then sup(B) = sup(A)2. a. Prove the claim. b. Does the claim still hold if we replace R+ with R? Explain briefly. So I have spent the past hours trying to prove this claim using the...
  2. J

    MHB Real Analysis - Prove the Riemann Integral Converges

    Just a couple questions. Problem 2: Just would like to know if this is the correct approach for this problem. Problem 3: I am just wondering if I can use Problem 2 to prove the first part of Problem 3? Because to me, they seem very similar. Problem 4: Would I use the MVT for integrals...
  3. J

    MHB Real Analysis - Riemann Integral Proof

    I have no idea how to incorporate the limit into the basic definitions for a Riemann integral? All we have learned so far is how to define a Riemann integral and the properties of Riemann integrals. What should I be using for this?
  4. PsychonautQQ

    The Subsequential Limit Points of a Bounded Sequence

    Homework Statement Let (a_n) be a bounded sequence. Prove that the set of subsequential limit points of (a_n) is a subsequentially compact set Homework Equations To be a subsequentutially compact set, every sequence in the set of limit points of (a_n) must have a convergent subsequence. The...
  5. M

    Studying Why do Walter Rudin's proofs in real analysis often seem so elusive and clever?

    Dear all, I currently a student in mechanical engineering and i reached the conclusion that maths from the point of view of mathematicians is lot more interesting than the eyes of engineers (for me at least). One of my friends in the maths department suggested to me to read real...
  6. kwangiyu

    Show that ##\lim_{n->\infty} \frac{n^2}{2^n} = 0 ##

    Homework Statement show that \lim_{n->\infty} \frac{n^2}{2^n} = 0 Homework Equations squeeze theorem The Attempt at a Solution I tried to use squeez theorem. I don't know how to do it because don't know how to reduce 2^n However, I can solve this question like this. Given \epsilon>0...
  7. F

    I Proof that lattice points can't form an equilateral triangle

    From Courant's Differential and Integral Calculus p.13, In an ordinary system of rectangular co-ordinates, the points for which both co-ordinates are integers are called lattice points. Prove that a triangle whose vertices are lattice points cannot be equilateral. Proof: Let ##A=(0,0)...
  8. A

    I How Does Real Analysis Justify Manipulation of Differential Elements in Physics?

    Suppose I wanted to prove the work-kinetic energy theorem. This means that I want to show that \frac{1}{2}m( \vec {v}^2_f - \vec{v}^2_i)=\int_{x_1}^{x_2} \vec{F} \cdot dx. So, I go ahead and start on the right side: \int_{x_1}^{x_2} (m \frac{d\vec{v}}{dt}) \cdot dx = m \int_{x_1}^{x_2}...
  9. B

    Range of a weird function

    Homework Statement Find the range ##y = \sqrt{\ln({\cos(\sin (x)}))}## Homework EquationsThe Attempt at a Solution [/B] https://www.desmos.com/calculator I used a graphing calculator to find the intersection between ##y = e^{x^2}## and ##y = \cos(\sin(x))##. Which I get as ##(0,1)##. So the...
  10. Josep

    Courses Should I retake Real Analysis I?

    Hi all, I am currently in my first semester of my sophomore year, taking Real Analysis I. This class covers formal proofs, properties of the real line, sequences, series, limits, continuity and differentiation, and Riemann Integration. I apparently got stuck with the worst professor at my...
  11. D

    Lower Bound on Weighted Sum of Auto Correlation

    Homework Statement Given ##v = {\left\{ {v}_{i} \right\}}_{i = 1}^{\infty}## and defining ## {v}_{n}^{\left( k \right)} = {v}_{n - k} ## (Shifting Operator). Prove that there exist ## \alpha > 0 ## such that $$ \sum_{k = - \infty}^{\infty} {2}^{- \left| k \right|} \left \langle {v}^{\left (...
  12. B

    B Why are these relations reflexive/symmetric/transitive?

    The definition of these relations as given in my textbook are : (1):- Reflexive :- A relation ##R : A \to A## is called reflexive if ##(a, a) \in R, \color{red}{\forall} a \in A## (2):- Symmetric :- A relation ##R : A \to A## is called symmetric if ##(a_1, a_2) \in R \implies (a_2, a_1) \in R...
  13. B

    B Flaw in my proof of something impossible

    Given :- $$g(f(x_1)) = g(f(x_2)) \implies x_1 = x_2$$ Question :- Check whether ##g(x)## is injective or not. Now this is of-course false; counter examples are easy to provide. But I proved that ##g(x)## must be one-one even after knowing the fact it must not. Here is the proof :- Let...
  14. Saph

    Studying How to study from a Real Analysis textbook like this

    Hello, I am taking a class in RA, where we're using Bartle/Sherbert. Since I have studied few chapters from it in the summer before, I decided to take a look at a more rigorous book, like baby rudin, but since many have advised against that book, I turned to Pugh's real mathematical analysis...
  15. jaskamiin

    Analysis Have a math degree, need to refamiliarize with advanced math

    I've been out of school for a while and working as a programmer. I want to start taking some masters courses for applied math (PDEs, numerical analysis, etc) and need to become familiar again with the advanced math I used to use in undergrad. I took two semesters of real analysis as an...
  16. FeDeX_LaTeX

    I Discrete Convolution of Continuous Fourier Coefficients

    Suppose that we have a 2\pi-periodic, integrable function f: \mathbb{R} \rightarrow \mathbb{R}, whose continuous Fourier coefficients \hat{f} are known. The convolution theorem tells us that: $$\displaystyle \widehat{{f^2}} = \widehat{f \cdot f} = \hat{f} \ast \hat{f},$$ where \ast denotes the...
  17. B

    Analysis Readability of Rudin's Real and Complex Analysis

    So I decide to self-study the real analysis (measure theory, Banach space, etc.). Surprisingly, I found that Rudin-RCA is quite readable; it is less terse than his PMA. Although the required text for my introductory analysis course was PMA, I mostly studied from Hairer/Wanner's Analysis by Its...
  18. DavideGenoa

    I Substitution in a Lebesgue integral

    Hi, friends! I read that, if ##f\in L^1[c,d]## is a Lebesgue summable function on ##[a,b]## and ##g:[a,b]\to[c,d]## is a differomorphism (would it be enough for ##g## to be invertible and such that ##g\in C^1[a,b]## and ##g^{-1}\in C^1[a,b]##, then...
  19. A

    Prove f(y) = y for every real number y

    A function f: R->R is a continuous function such that f(q) = q for every rational number q. Prove f(y) = y for every real number y. I know every irrational number is the limit to a sequence of rational numbers. But I not sure how to prove f(y) = y for every real number y. Any ideas?
  20. DavideGenoa

    I Relaxed conditions for the density: Ampère's law still valid?

    The most common proof that I have found of the fact that Ampère's law is entailed by the Biot-Savart law essentially uses the fact that, if ##\boldsymbol{J}:\mathbb{R}^3\to\mathbb{R}^3##, ##\boldsymbol{J}\in C_c^2(\mathbb{R}^3)##, is a compactly supported twice continuously differentiable field...
  21. R

    Prove Continuous Functions Homework: T Integral from c to d

    Homework Statement Prove $$T\int_c^d f(x,y)dy = \int_{c}^dTf(x,y)dy$$ where $$T:\mathcal{C}[a,b] \to \mathcal{C}[a,b]$$ is linear and continuous in L^1 norm on the set of continuous functions on [a,b] and $$f:[a,b]\times [c,d]$$ is continuous. Homework EquationsThe Attempt at a Solution [/B]...
  22. K

    Schools University Mathematics Abstraction

    I'm currently a first year MathPhys student, and next year I have to decide my stream. I can pick a combination (pure) Mathematics, Applied & Computationtal. Mathematics, Statistics, MathSci, Physics, Theoretical Physics or Physics with Astronomy & Space. Naturally there are restrictions, and I...
  23. W

    I Supremum and Infimum Proof

    Let S and T be subsets of R such that s < t for each s ∈ S and each t ∈ T. Prove carefully that sup S ≤ inf T. Attempt: I start by using the definition for supremum and infinum, and let sup(S)= a and inf(T)= b i know that a> s and b< t for all s and t. How do i continue? , do i prove it...
  24. SrVishi

    Other Proof Tips for Math Majors: Logic & Techniques for Real Analysis

    Every math major eventually learns logic and standard proof techniques. For example, to show that a rigorous statement P implies statement Q, we suppose the statement P is true and use that to show Q is true. This, along with the other general proof techniques are very broad. A math major would...
  25. M

    Intuitive explanation of lim sup of sequence of sets

    Hi, I can derive a few properties of the limit inferior and limit superior of a sequence of sets but I have trouble in understanding what they actually mean. However, my understand of lim inf and lim sup of a sequence isn't all that bad. Is there a way to understand them intuitively (something...
  26. B

    Analysis Seeking a Rudin's PMA-level analysis book with abstract proofs

    Dear Physics Forum personnel, I recently got interested in the art of abstract proof, where the focus is writing the proof as general as possible rather than starting with a specific cases. Could anyone recommend an analysis book at the level of Rudin's PMA that treats the introductory...
  27. samgrace

    Calculus Derivations: Handbook, Rules, Properties & Books

    Hello, Please take a look at this handbook of derivatives and integrals: http://myhandbook.info/form_diff.html http://integral-table.com/downloads/single-page-integral-table.pdf I would appreciate it if someone could point me in the direction of exemplary books that derive these...
  28. P

    Analysis Good supplementary real analysis book

    So the course I'm taking doesn't have a textbook requirement just lecture notes as the study material. While these are sufficient I would like to supplement with an outside reference that is a bit more in depth / explanatory. It's your typical undergrad real analysis course covering: The least...
  29. B

    Analysis Which book will suit the following course syllabus (introductory analysis)?

    Dear Physics Forum personnel, I am a undergraduate student with math and CS major who is currently taking an introductory analysis course called MATH 521 (Rudin-PMA). On the next semester, I will be taking the course called MATH 522, which is a sequel to 521. My impression is that 522 will be...
  30. B

    Analysis What are the best introductory books for real analysis?

    Dear Physics Forum personnel, I am a college student with huge enthusiasm to the analysis and theoretical computer science. In order to start my journey to the real analysis. I am currently taking an introductory-analysis course (Rudin-PMA; I also use Shilov too) and linear algebra...
  31. T

    Proving Compact Set Exists with m(E)=c

    Homework Statement Suppose E1 and E2 are a pair of compact sets in Rd with E1 ⊆ E2, and let a = m(E1) and b=m(E2). Prove that for any c with a<c<b, there is a compact set E withE1 ⊆E⊆E2 and m(E) = c. Homework Equations m(E) is ofcourese referring to the outer measure of E The Attempt at a...
  32. T

    Analysis Answers to questions from the book: Real Analysis by Stein

    Hi I am trying to teach myself Measure Theory and I am using the book: Real Analysis by Stein and Skakarchi from Princeton. I want to check if my answers to the questions are correct, so I am asking: Does anyone have the answers to the questions in chapter 1 ?
  33. Z

    Proof: Every convergent sequence is Cauchy

    Hi, I am trying to prove that every convergent sequence is Cauchy - just wanted to see if my reasoning is valid and that the proof is correct. Thanks! 1. Homework Statement Prove that every convergent sequence is Cauchy Homework Equations / Theorems[/B] Theorem 1: Every convergent set is...
  34. Andreol263

    Analysis A good real analysis introductory book

    Hi guys, my first question is:what i really need to understand real analysis? and the second is on the title:could some of you recommend a good book on real analysis? cause I've found some texts that are very difficult to understand some concepts...
  35. Z

    Interior Point of 1/n

    Hi All, A simple question but one for which I cannot seem to get the intuition. 1. Homework Statement Find the interior point of {1/n : n ∈ ℕ}. Homework Equations N/A The Attempt at a Solution Let S = {1/n : n ∈ ℕ}, where S ⊆ℝ x is an interior point if ∃N(x ; ε), N(x ; ε) ⊆ S. My...
  36. Z

    Real Analysis - Infimum and Supremum Proof

    Hi Guys, I am self teaching myself analysis after a long period off. I have done the following proof but was hoping more experienced / adept mathematicians could look over it and see if it made sense. Homework Statement Question: Suppose D is a non empty set and that f: D → ℝ and g: D →ℝ. If...
  37. infinite.curve

    Resources For Real Analysis and Concepts of Mathmatics

    I have been browsing the web, and I notice that I could not find any websites that have real analysis text around. Yes, I understand that I should look for books written by professionals in the field, but I do not know which one I should buy. Do you know of some online resources to real analysis...
  38. M

    Proving a sequence converges

    Homework Statement : [/B]Prove that if xn is a sequence such that |xn - xn+1| ≤ (1/3n), for all n = 1,2,..., then it converges.Homework Equations : [/B]The definition of convergence.The Attempt at a Solution :[/B] I attempted to prove this by induction, so I am clearly far off the mark here...
  39. M

    Showing a sequence converges to its supremum

    Homework Statement : [/B]Let a = sup S. Show that there is a sequence x1, x2, ... ∈ S such that xn converges to a.Homework Equations : [/B]I know the definition of a supremum and convergence but how do I utilize these together?The Attempt at a Solution :[/B] Given a = sup S. We know that a =...
  40. V

    Interested in Joining My Polymath Project on Real Analysis?

    I forgot to formally introduce myself on this forum. I am VKnopp. I am 14 year old maths enthusiast with Asperger's Syndrome. I self-educated myself all the way up to Calculus III with a little bit of number theory, linear algebra, complex analysis and real analysis supplements. I am in the top...
  41. L

    How to prove the following defined metric space is separable

    Let ##\mathbb{X}## be the set of all sequences in ##\mathbb{R}## that converge to ##0##. For any sequences ##\{x_n\},\{y_n\}\in\mathbb{X}##, define the metric ##d(\{x_n\},\{y_n\})=\sup_{n}{|x_n−y_n|}##. Show the metric space ##(\mathbb{X},d)## is separable. I understand that I perhaps need to...
  42. R

    Summer Upper Level Math Courses Online?

    Econ Major here. I plan to graduate in spring 2016 and from there apply to economics grad programs. I still need to take Advanced Math and Advanced Calculus, and Real Analysis, all of which are not available during the summer at my uni (Florida International University). Anyone know of any...
  43. K

    Foundations Theoretical Books on Mathematics

    What are some rigorous theoretical books on mathematics for each branch of it? I have devised a fantastic list of my own and would like to hear your sentiments too. Elementary Algebra: Gelfand's Algebra Gelfand's Functions & Graphs Burnside's Theory of Equations Euler's Analysis of the...
  44. B

    Taking Real Analysis, Abstract Algebra, and Linear Algebra

    Dear Physics Forum advisers, I am a college sophomore in US with a major in mathematics, and an aspiring algebraic number theorist and cryptographer. I wrote this email to seek your advice about taking the Analysis I (Real Analysis I), Abstract Algebra I, and Linear Algebra with Proofs. At...
  45. QuantumCurt

    Complex before real analysis? How's my fall schedule look?

    Hey everyone, I'm transferring into UIUC this fall, and I just registered for my classes earlier today. I'm completing dual degrees in physics and math. I've completed the introductory physics sequence, and the introductory calculus sequence, plus a 200 level introductory differential equations...
  46. B

    High Resolution Upper Division Undergrad Math lecture videos

    I'm interested in watching videos of Real Analysis lectures etc. in good quality resolution. Those Harvey Mudd College lectures are valuable but annoying re video quality. Thanks. - Blue
  47. Z

    How Does the Invariance Principle Apply to Limits in Engel's Problem?

    Homework Statement Hi Guys, This is the first exampe from Engel's problem solving book. After a long period of no math I am self studying. I do not know where my knowledge deficits lie, and was recommended this site for help. "E1. Starting with a point S (a, b) of the plane with 0 < b < a...
  48. A

    Proving f = 0 almost everywhere

    I am working on a problem##^{(1)}## in Measure & Integration (chapter on Product Measures) like this: Suppose that ##f## is real-valued and integrable with respect to 2-dimensional Lebesgue measure on ##[0, 1]^2## and also ##\int_{0}^{a} \int_{0}^{b} f(x, y) dy dx = 0## for all ##a, b \in...
  49. F

    Application of Lebesgue differentiation theorem

    If ## f\in L_{p}^{\rm loc}(\mathbb{R}^{n}) ## and ## 1\leq p<\infty ##, then a stronger version of Lebesgue differentiation theorem holds: $$\lim\limits_{r\rightarrow 0}\dfrac{\|f\chi_{B(x,r)}\|_{L_{p}(\mathbb{R}^{n})}}{\|\chi_{B(x,r)}\|_{L_{p}(\mathbb{R}^{n})}}=|f(x)|$$ for almost all ##...
  50. DrPapper

    Exploring Mary Boas' Theorem III: Analytic Functions & Taylor Series

    On page 671 Mary Boas has her Theorem III for that chapter. Roughly it tells us that if f(z) -a complex function- is analytic in a region, inside that region f(z) has derivatives of all orders. We can also expand this function in a taylor series. I get the part about a Taylor series, that's...
Back
Top