What is Real analysis: Definition and 511 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. 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...
  2. 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 *...
  3. S

    Is real analysis necessary for success in statistics?

    Hello there, Is real analysis really important in the process of learning statistics? The reason is, I suck at real analysis(actually, I failed it) and do really good at stats(I am in my third yr. at uni.). Should I continue studying stats or switch to another major? I am looking...
  4. W

    Is differential equation required to study real analysis?

    Hi all, Is differential equation a prerequisite to study real analysis (in context of baby Rudin)? And does it have any use in measure theory or Stochastic Calculus? Thanks in advance.
  5. W

    Is differential equation required to study real analysis?

    Hi all, Is differential equation a prerequisite to study real analysis (in context of baby Rudin)? And does it have any use in measure theory or Stochastic Calculus? Thanks in advance.
  6. T

    Real Analysis: Proving an*bn converges to ab

    Homework Statement Use the fact that an= a + (an - a) and bn= b + (bn - b) to establish the equality an*bn - ab = (an-a)(bn-b)+b(an-a)+a(bn-b). Then, use this equality to prove that the sequence {an*bn} converges to ab. Homework Equations Definition of convergence: |an*bn - ab| < ε...
  7. H

    Prove or disprove: Justify your answer using the def or limit. Real analysis

    Homework Statement Suppose f: ℝ-{0} → ℝ has a positive limit L at zero. Then there exists m>0 such that if 0<|x|<m, then f(x)>0.Homework Equations The definition of the limit of a function at a point is: (already assuming f to be a function and c being a cluster point) A real number L is said...
  8. C

    Feelings discouraged by Real Analysis

    (typo: title should be Feeling, not Feelings. Whoops) Hi there, I'm a second-year student at a highly ranked private liberal arts school and I am pursuing a BA in Math. Once I graduate, I want to pursue a PhD in either Math or CS. I obviously don't know exactly what I want to focus on, but...
  9. T

    Real Analysis: countably infinite subsets of infinite sets proof

    Homework Statement Prove that every infinite subset contains a countably infinite subset. Homework Equations The Attempt at a Solution Right now, I'm working on a proof by cases. Let S be an infinite subset. Case 1: If S is countably infinite, because the set S is a subset...
  10. T

    Real Analysis: one to one correspondence between two countably infinite sets

    Homework Statement Suppose that A and B are both countably infinite sets. Prove that there is a one to one correspondence between A and B. Homework Equations The Attempt at a Solution By definition of countably infinite, there is a one to one correspondence between Z+ and A and...
  11. C

    Good real analysis books recommendations please.

    Hello, I'm not quite sure if this kind of question can be posted on this board. Please excuse me if not. I started studying real analysis with Rudin's Principles of Mathematics which was relatively compact. Then I bought Apostol's book which was much more helpful because it was more...
  12. S

    Courses Real analysis and Integration course for EE

    Hi, I'm an undergraduate 4th year Electronics engineering student. So far I have taken courses from various fields of microelectronics and telecommunications. This year, I've decided to direct my career more to telecommunications (might be a field like wireless communications, digital image...
  13. T

    Real Analysis Supremum of a Set Proof

    Homework Statement Let S be a non empty set that is bounded about and β = sup S. Prove that for each ε > 0 there exists a point x in S such that x > β - ε.Homework Equations The Attempt at a Solution I don't really know how to begin this. I know it's true; I'm looking at the problem and I'm...
  14. N

    Top Real Analysis Books for Undergraduates: Expert Recommendations

    Hi, I am currently taking real analysis(undergraduate course), and am using the book "Introduction to Real Analysis" by Bartle and Sherbert. I think the book is okay overall, but I was hoping to purchase a secondary text to look at whenever I am confused with a proof or something in the...
  15. B

    Proving the Existence of m for (1-1/m)^n > 1-ε in Real Analysis

    Let n ≥ 1 be an integer and ε > 0 a real number. Without making reference or use of nth roots, prove that there exists a positive integer m such that \left (1- \frac{1}{m} \right )^{n}> 1-\varepsilon How would I go about proving this? Would I just solve for m?
  16. S

    Real Analysis Question: Sequences and Closed Sets

    Homework Statement Let {xn} be a sequence of real numbers. Let E denote the set of all numbers z that have the property that there exists a subsequence {xnk} convergent to z. Show that E is closed. Homework Equations A closed set must contain all of its accumulation points. Sets with no...
  17. P

    Radius of convergence: 1/(1+x^2) about 1, using only real analysis

    I've seen this thread: https://www.physicsforums.com/showthread.php?t=297842 and that is the exact question I need to to answer. What is the radius of convergence of 1/(1+x^2) expanded about x_0=1? The problem is, I can only use an argument in real analysis. I see the answer is...
  18. F

    Real analysis - differential/sequence

    Homework Statement I attached a .bmp file Homework Equations The Attempt at a Solution I don't get this one at all. Since f is differentiable, the interval [a,b] must be continuous, and I cannot use rational/irrational tricks like in limit problems. does anyone have suggestions?
  19. S

    Real Analysis Riemann Integration

    Suppose we have: f(x)= 1 if 0\leq x \leq 1 AND 2 if 1\leq x \leq 2 Using the definition, show that f is Riemann integrable on [0, 2] and find its value? I have a general idea of how to complete this question using partitions and the L(f,P) U(f,P) definition, but am not quite receiving the...
  20. N

    Real analysis: show that a continuous function is defined for irrationals

    Let f be a continuous function defined on (a, b). Supposed f(x)=0 for all rational numbers x in (a, b). Prove that f(x)=0 on (a, b). i don't even know where to start...any tips just to point me in the right direction?
  21. F

    Real analysis: Limit of a product of sequences

    Homework Statement Let (u_{n})_{n} be a real sequence such that lim u_{n} = 0 as x→∞ and let (v_{n})_{n} be a bounded sequence. Show that lim (u_{n})_{n}(v_{n})_{n} = 0 as x→∞ Homework Equations The Attempt at a Solution Since (v_{n})_{n} is bounded then it has a least upper bound and...
  22. H

    Real Analysis 2 a year after Real Analysis 1?

    I'm currently attending university, and I'm comfortable with remembering my math skills. However, I am planning to take Real Analysis 1 in the Spring '12 and then the 2nd course in Spring '13. During that time, I will be taking Advanced Calc, but what do you think? Do you think this will...
  23. F

    Real analysis question: show that x^4 - x - 1 = 0 has two real roots

    Homework Statement Let f(x)=x^4 - x - 1. Show that f(x)=0 has two real roots. Homework Equations None The Attempt at a Solution x(x^3 - 1 - 1/x) = 0 which gives x=0 and x^3 - 1 - 1/x=0, x^2 - 1/x - 1/x^2=0, but WolframAlpha says x~~0.724492 and x~~-1.22074. I kept dividing by x it but...
  24. L

    Real Analysis: Continuity & Intervals

    Homework Statement If the domain of a continuous function is an interval, show that the image is an interval. Homework Equations Theorem from book: f is a cont. function with compact domain D, then f is bounded and there exists points y and z such that f(y) = sup{ f(x) | x ∈ D} and...
  25. A

    HELP real analysis question: continuity and compactness

    HELP! real analysis question: continuity and compactness Homework Statement Let (X,d) be a metric space, fix p ∈ X and define f : X → R by f (x) = d(p, x). Prove that f is continuous. Use this fact to give another proof of Proposition 1.126. Proposition 1.126. Let (X, d) be a metric space...
  26. S

    Linear Algebra and Real Analysis 1?

    Do you think it will be too much of a load to take Linear Algebra and Real Analysis in the same semester? Please note that this Linear Algebra course is not an intro course, it's an upper level mathematics course. Will it be similar to the intro to Linear Algebra course I'm taking now? Also, in...
  27. S

    Is real analysis really that hard?

    I'm a sophomore math major, and I' currently taking proofs, linear algebra (not proofs-based), and calc 3. These classes aren't that bad so far. I met with a math adviser today, and he told me for my major requirements I should take real analysis 1&2, Linear algebra, and abstract algebra for a...
  28. M

    Two real analysis problems: proving constancy and a uniform convergence problem

    The problem statement Let f:[a,b]→\mathbb{R} be differentiable and assume that f(a)=0 and \left|f'(x)\right|\leq A\left|f(x)\right|, x\in [a,b]. Show that f(x)=0,x\in [a,b]. The attempt at a solution It was hinted at that the solution was partly as follows. Let a \leq x_0 \leq b. For all x\in...
  29. Z

    What are the applications of real analysis?

    So, I am taking introduction to real analysis next semester, and I heard that it may be a challenging course. But what I want to know is the applications of real analysis, which I imagine there are some of, considering how the phrase "real analysis" seem to be thrown around quite a bit. Thanks.
  30. N

    Real Analysis: show sequences have the same limit if |Xn-Yn| approaches 0

    Homework Statement Suppose {Xn}, {Yn} are sequences in ℝ and that |Xn-Yn|→0. Show that either: a) {Xn} and {Yn} are both divergent or b) {Xn} and {Yn} have the same limit. Homework Equations N/A The Attempt at a Solution I first prove that lim(Xn-Yn)=lim(Xn)-lim(Yn). I am not...
  31. L

    Real Analysis problem (easy), Triangle inequality

    Homework Statement > a[1], a[2], a[3], .. , a[n] are arbitrary real numbers, prove that; abs(sum(a[i], i = 1 .. n)) <= sum(abs(a[i]), i = 1 .. n) Homework Equations The Attempt at a Solution I have uploaded my attempt as a pdf file, since I'm not too familiar with the...
  32. E

    Proof of adding powers (real analysis)

    Homework Statement Prove that br+s=brbs if r and s are rational. Homework Equations So far we know the basic field axioms and a a few other things related to powers. 1.) For every real x>0 and every integer n>0 there is one and only one positive real y such that yn=x 2.) if a and b...
  33. M

    How can I determine the linear part of f(A + H) in terms of H?

    Homework Statement Let f:Rnxn-->Rnxn be defined by f(A) = A2. Prove that f is differentiable. Find the derivative of f. Homework Equations f(a + h) = f(a) + f'(a)h + \phi(h) The Attempt at a Solution f(A + H) = (A + H)2 = A2 + AH + HA + H2 f(A) is given by A2. So the sum of...
  34. estro

    Real Analysis Study Group: Join Us in Exploring the Fundamentals of Analysis

    Anyone interested in opening online study group on Real Analysis? I want to use https://www.amazon.com/gp/product/0486469131/?tag=pfamazon01-20 for the study group. Method: Some time will be given for self study then, group will discuss concepts and solve exercises from the book. [each...
  35. P

    How Do You Prove That g Equals f^-1 Given Composition Identities?

    Let f and g be functions such that (g\circf)(x)=x for all x \epsilonD(f) and (f\circg)(y)=y for all y \epsilonD(g). Prove that a g = f^-1 Pf/ How would you go about starting this besides saying Let f and g be functions such that (g\circf)(x)=x for all x \epsilonD(f) and (f\circg)(y)=y for...
  36. D

    Intro to Real Analysis: Supremum

    Homework Statement Find the supremum of E=(0,1) Homework Equations The Attempt at a Solution By definition of open interval, x<1 for all x in E. So 1 is an upper bound. Let M be any upper bound. We must show 1<=M. Can I just say that any upper bound of M must be greater than or...
  37. D

    Proving a < b for All Positive Epsilon in Real Analysis

    Homework Statement If a < b-\epsilon for all \epsilon >0, then a<0 Homework Equations All I really have are the field axioms of the real numbers and the order axioms (trichotomoy, transitive, additive property, multiplication property). The Attempt at a Solution Well I broke this...
  38. N

    Proving the Inequality of Infimums for Bounded Functions on [0,1]

    I Just started Analysis 1 this week and I've encountered some tricky problems in the Assignment Homework Statement Let f,g : [0,1] -> R be bounded functions. Prove that inf{ f(x) + g(1-x) : x (element of) [0,1]} >= inf{f(x) : x (element of) [0,1]} + inf{g(x) : x (element of) [0,1]}...
  39. M

    Real Analysis: Proving the Greatest Lower Bound Property

    Homework Statement (a) Suppose that A and B are nonempty subsets of R. Define subsets -A={-x: x\inA} and A+B={x+y: x\inA and y\inB}. Show that if A and B are bounded above, then the greatest lower bound of -A = - least upper bound of A and the least upper bound of (A+B) = the least upper bound...
  40. W

    Proofs with epsilon delta (real analysis)

    Hello, I have stumbled upon a couple of proofs, but I can not seem to get an intuitive grasp on the what's and the whys in the steps of the proofs. Strictly logical I think I get it. Enough talk however. Number 1. "Let f be a continuous function on the real numbers. Then the set {x in R ...
  41. S

    Basic knowledge about real analysis

    members i need some basic knowledge about real analysis i got lot of trouble... about this topic
  42. A

    What Are the Best Supplementary Materials for a Beginner's Real Analysis Course?

    I'm likely taking an introductory real analysis course in the fall, and I was wondering what supplementary material I should look into. I'm working my way through Velleman's proofs book, what else would you recommend as a supplement to a first course in RA?
  43. D

    Studying What is the point of studying real analysis?

    What is the point of studying real analysis? I find it not to be very useful...
  44. A

    Is A+B Closed for Two Closed Sets in Real Analysis?

    Homework Statement Suppose A is a compact set and B is a closed subset of Rk. then A+B is closed in Rk. show that A+B for two closed sets is not necessarily closed by a counter-example. well, since A is a compact set and there's a theorem in Rudin's mathematical analysis chapter 2 stating...
  45. A

    A very open ended question about Real Analysis

    I was hoping to get some personal opinions regarding the first round (two semester sequence) of undergraduate analysis. How difficult do YOU think that these classes are? Use comparisons as you feel fit (linear algebra, intro proofs course, abstract algebra, etc). (I do realize how...
  46. A

    What Real Analysis book do you suggest?

    Hi all, I've been self-studying Rudin's Mathematical Analysis recently and have studied the first 4 chapters so far and I'm fine with the way it has developed the theory but the book lacks solved exercises and examples to be called a perfect book for self-studying. I have learned the general...
  47. A

    After a Real Analysis book that has solutions (for self-study)

    One that is suitable for self-study and doesn't require me to constantly ask the internet for clarifications. 'Understanding Analysis' by Stephen Abbott and 'Real Mathematical Analysis' by C.C. Pugh seem suitable but unfortunately I can't find a solutions manual Thanks EDIT: Also I need a...
  48. A

    Which is more helpful for real analysis?

    Please note that this is a "double post". I was not sure if I should put this here or in the calculus and analysis subform. If you must delete, I understand. But please, delete the one that should actually be deleted. Thanks, and sorry =|...
  49. K

    Linear Algebra and Real Analysis Review

    I'm going into my 3rd year as a pure math major at UWO. I have completed both second year Real Analysis and Linear Algebra with decent marks. However, I really feel that I didn't take too much from both other than the general concepts, especially second semester of Linear Algebra (due to...
  50. T

    Real Analysis / Advanced Calc Puzzler

    Let f:[a,b] \rightarrow R be a continuous function such that f(a)=f(b)=0 and f' exists on (a,b). Prove that for every real \lambda there is a c \in (a,b) such that f'(c) = \lambda f(c).
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