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. N

    Starting Postgraduate Programme in Economics, Advice for Calculus - Real Analysis

    Hello, I am starting a postgraduate level Economics course in two months. I will have to go through some kind of a Math Camp before the course, lasting more or less 10 days. Here is my curriculum; 1. REAL ANALYSIS Topics: • Sequences and Convergence • Function on Rn • Continuity •...
  2. A

    Exploring Empty Functions in Real Analysis

    I'm currently reading Sterling Berberian's Foundations of Real Analysis, and the first chapter had an overview of foundational mathematics from axiomatic set theory to constructive proof of the real numbers. I was looking over this chapter, and I found this exercise in the functions section...
  3. D

    Regarding limits in Real Analysis

    Question: Suppose that f(x)>0 on (0,1) and that lim as x goes to 0 exists for the function. Show that lim as x goes to 0 for the function is greater than or equal to 0. So I know that intuitively that this is true for obvious reasons, but I can not think of a clever way to set up the proof...
  4. D

    Real Analysis: Continuity and Uniform Continuity

    Question: Show that f(x)= (x^2)/((x^2)+1) is continuous on [0,infinity). Is it uniformly continuous? My attempt: So I know that continuity is defined as "given any Epsilon, and for all x contained in A, there exists delta >0 such that if y is contained in A and abs(y-x)<delta, then...
  5. E

    Applicability of Intro To Algebra and Intro to Real Analysis to Physics

    Applicability of "Intro To Algebra" and "Intro to Real Analysis" to Physics Well, due to timetable complications I'm having to search for courses that aren't apart of my graduation requirements so I'm thinking about taking some math courses. Which one of these courses do you think is more...
  6. K

    Real analysis: inequality limitsuperior/inferior

    Homework Statement Consider \sum_{1}^{\infty} a_{n}, a_{n} \neq 0 Show that \underline{\lim\limits_{n \rightarrow \infty}}|\frac{a_{n+1}}{a_{n}}| \leq \underline{\lim\limits_{n \rightarrow \infty}}\sqrt[n]{|a_{n}|}\leq \overline{\lim\limits_{n \rightarrow \infty}}\sqrt[n]{|a_{n}|)}...
  7. I

    Equivalence of 8 properties in Real Analysis

    Please help me prove that the following properties are equivalent Nested Interval Property Bolzano-Wierstrass theorem Monotonic sequence property LUB property Heine-Borel theorem archimedean property and cauchy sequence line connectedness...
  8. M

    Closed Subset Addition in Metric Spaces: Real Analysis Homework Help

    Homework Statement Let E, F be two closed and non-empty subsets of R, where R is seen as a metric space with teh distance d(a,b)=|a-b| for a,b ϵ R. Suppose E + F := { e+f |e ϵ E, f ϵ F}. Is is true that E+F has to be closed? Homework Equations The Attempt at a Solution I'm...
  9. H

    Real Analysis. Prove f(x) = logx given all these conditions.

    It's just the final part (e) that I don't get, I have the rest but just for completeness I thought I'd put it in (iii) Let f : (0,infinity) -> R be a function which is differentiable at 1 with f '(1) = 1 and satisfies: f(xy) = f(x) + f(y) (*) (a) Use (*) to determine f(1) and show that f(1/x)...
  10. Demon117

    Understanding Zero Sets: Real Analysis Examples

    What is the definition of a zero set and what exactly does it mean? I have come across different responses on the internet, but none of them explain really what it means or give good examples, I am having a rough time with this concept in real analysis. For example, how would I determine...
  11. M

    Real Analysis proof (inner product)

    Hello all, I am having trouble showing that the operation defined by f*g(f of g)= Integral[from a to b]f(x)g(x) is an inner product. I know it must fulfill the inner product properties, which are: x*x>=0 for all x in V x*x=0 iff x=0 x*y=y*x for all x,y in V x(y+z)=x*z+y*z...
  12. P

    Real Analysis (Cantors Diagonalisation?)

    Homework Statement Let S be the set of all functions u: N -> {0,1,2} Describe a set of countable functions from S Homework Equations We're given that v1(n) = 1, if n = 1 and 2, if n =/= 1 The function above is piecewise, except i fail with latex The Attempt at a Solution...
  13. E

    Studying Real Analysis textbook and study guide

    Hey, I'm taking a bit of a flyer here, but does anyone know of a half decent online textbook that also has a study guide? My class is working out of Trench's online book, but more or less it's just for a reference and problems, we mainly just work from notes. Would there be any book (online)...
  14. M

    Real analysis: limit of sequences question

    ok so, a) If s sub n→0, then for every ε>0 there exists N∈ℝ such that n>N implies s sub n<ε. This a true or false problem. Now this looks like a basic definition of a limit because s sub n -0=s sub n which is less than epsilon. n is in the natural numbers. But, I thought there should be...
  15. E

    Real Analysis convergence proof

    Homework Statement If the sequence xn ->a , and the sequence yn -> b , then xn - yn -> a - b The Attempt at a Solution Can someone check this proof? I'm aware you cannot subtract inequalities, but I tried to get around that where I indicated with the ** in the following proof...
  16. B

    Metric Spaces: Proving d is a Metric & Defining \widetilde{} on Cauchy Sequences

    Question A: Let (xn) and (yn) are two Cauchy sequences in a metric space (X, d), define d((xn), (yn)) = lim d(xn, yn). It is easy to prove that "d" is a metric on the set of all Cauchy Sequences. Now let's define \widetilde{} on the set of all sequences in a metric space (X, d) by (xn)...
  17. P

    Real Analysis, Lebesgue, limit of an integral

    I am absolutely lost. I had to take Advanced Calculus as independent study in a one month class and this book has very few examples, if any. I'm not even sure where to start on this one. I have to compute the limit of an integral and then justify my methods according to the Lebesgue theory...
  18. T

    Constructing a Sequence with Given Limit Points

    Homework Statement Let {y_j} be N given real numbers. Construct a sequence {a_n} so that {y_j} is the set of limit points of {a_n}, but a_n ≠ y_j for any n or j. Homework Equations Bolzano-Weierstrass theorem The Attempt at a Solution Have no idea how to go about it. I'd really...
  19. E

    Help with limit proofs for real analysis

    I'm not quite sure if this is the correct subforum. I was wondering if anybody knew where I could find some decent real analysis notes or lectures online, specifically on the formal definition of a limit. My prof is great, I just missed the class and the textbook and notes aren't quite making...
  20. H

    Need a lot of worked real analysis proofs (from easy to difficult)

    I was accepted into a top tier Ph.D. Operations Research program. I have six months to prepare independently on my own (at home). Everybody told me real analysis is the first thing I should look at (which makes sense, because I don't have proof experience). Can you please recommend me a book...
  21. R

    Real Analysis - Simple supremum/infimum problem

    Homework Statement If S = { 1/n - 1/m | n, m \in N}, find inf(S) and sup(S) I'm having a really hard time wrapping my head around the proper way to tackle sumpremum and infimum problems. I've included the little that I've done thus far, I just need a nudge in the right direction. Correct me...
  22. I

    Why Is Real Analysis Critical in Science and Engineering?

    i like limit, continuity,differentiation in real analysis, they are interesting, but i don't know what is their importance? And about lebesgue integration, i don't think it is interesting, and it seems it is useless
  23. C

    How Do You Prove Inequality for Bounded Functions in Real Analysis?

    Homework Statement Let f and g be bounded functions on [a,b]. 1. Prove that U(f+g)</=U(f)+U(g). 2. Find an example to show that a strict inequality may hold in part 1. Homework Equations Definition of absolute value? The Attempt at a Solution I know that a function f is bounded if its...
  24. R

    Real Analysis: Proving Lim(yn)=0 from Lim(xn)=Infinity & Lim(xnyn)=L

    Question : Let (xn) and (yn) be sequences of real numbers such that lim(xn)= infinity and lim(xnyn)=L for some real number L. Prove Lim(yn)=0. I've been trying to solve this question for a long time now. I've no success yet. Can anyone guide me as to how i can approach it.
  25. A

    Proving Rudin Theorem 7.17: Real Analysis

    i am asked to prove the remark Rudin made in theorem 7.17 in his Mathematical Analysis. Suppose {fn} is a sequence of functions, differentiable on [a,b] such that {fn(x0)} converges for some x0 in [a,b]. Assume f'n (derivative of fn) is continuous for every n. Show if {f'n} converges...
  26. K

    Inequality question from Real Analysis

    Homework Statement let n\inN To prove the following inequality na^{n-1}(b-a) < b^{n} - a^{n} < nb^{n-1}(b-a) 0<a<b Homework Equations The Attempt at a Solution Knowing that b^n - a^n = (b-a)(b^(n-1) + ab^(n-2) + ... + ba^(n-2) + a^(n-1) we can divide out (b-a) because b-a #...
  27. Z

    Need help in a real analysis question

    I am trying to prove a question : Assume K\inR^{m} is compact and {xn} (n from 1 to infinite) is a sequence of points in K that does not converge . Prove that there are 2 subsequences that converge to different points in K . Hint : Let yi=x_{ni} be one subsequence that converges to a point in...
  28. U

    Real Analysis: Proof of convergence

    Homework Statement Prove if {bn} converges to B and B ≠ 0 and bn ≠ 0 for all n, then there is M>0 such that |bn|≥M for for all n. Homework Equations What I have so far: I know that if {bn} converges to B and B ≠ 0 then their is a positive real number M and a positive integer N such...
  29. U

    Real Analysis: product of convergent sequences

    Homework Statement suppose {an} and {bn} are sequences such that {an} converges to A where A does not equal zero and {(an)(bn)} converges. prove that {bn} converges. Homework Equations What i have so far: (Note:let E be epsilon) i know that if {an} converges to A and {bn}converges...
  30. K

    Real Analysis proof limits and bounded functions

    Homework Statement Let f be a function and p\in . Assume that a\leqf(x)\leqb near p. Prove that if L= lim f(x) as x-->p Then L\in [a,b] The Attempt at a Solution I want to say that because f(x) is bounded by [a,b] that automatically implies that the Limit L is also bounded by...
  31. K

    Real Analysis proof continuity

    Show that the function f(x)=x is continuous at every point p. Here's what I think but not sure if i can make one assumption. Let \epsilon>0 and let \delta=\epsilon such that for every x\in\Re |x-p|<\delta=\epsilon. Now x=f(x) and p=f(p) so we have |f(x)-f(p)|<\epsilon...
  32. K

    Help with Real analysis proof about limit laws and functions

    Homework Statement Let f be a function let p /in R. Assume limx->p=L and L>0. Prove f(x)>L/2 The Attempt at a Solution Let f be a function let p /in R. Given that limx->pf(x)=L and L>0. Since L\neq0 Let \epsilon= |L|/2. Then given any \delta>0 and let p=0 we have |f(x)-L| = |0-L| =...
  33. C

    What Can I Take Without Real Analysis?

    Currently, I am a bioengineering major, but I have been taking math electives the past year and a half, and now I am finding myself liking pure mathematics much more than engineering and only two classes away from a degree. The two courses I need are Real Analysis and Abstract Algebra...
  34. T

    Math Real Analysis Problem, Riemann Sum Integral?

    Part 1. Homework Statement The problem literally states... " The Integral. limit of n-> infinity of n*[1/(n+1)^2 + 1/(n+2)^2 + 1/(n+3)^2 + 1/(2n)^2] = 1/2 " According to the teacher, the answer is 1/2. I don't know why or how to get there. Part 2. The attempt at a solution...
  35. K

    Real Analysis proof Using definition that f is defined near p

    Let (a, b) be an open interval in R, and p a point of (a, b). Let f be a real-valued function defined on all of (a, b) except possibly at p. We then say that the limit of f as x approaches p is L if and only if, for every real ε > 0 there exists a real δ > 0 such that 0 < | x − p | < δ and x ∈...
  36. B

    Real Analysis Problem involving Image of a Bounded Set

    Homework Statement Ok so I'm given that we have some function from R to R, that is continuous on all of R. I am asked if it is possible to find some BOUNDED subset of R such that the image of the set is not bounded. The professor gave the hint: look at closures. Homework Equations...
  37. K

    Real Analysis Exam Questions: Need Help Studying!

    Real Analysis Exam Questions. Please Help! I'm taking this course on real analysis and my exam will be in less than a week from now :eek: These are exam questions from previous year which have been assigned as homework, and I just started studying and it's really hard. I would be sooo happy if...
  38. C

    Taking Calculus III and Real Analysis During Same Semester

    Is it wise to take Calculus III and Intro to Real Analysis during the same semester? Or should I complete Calculus III and take Intro to Real Analysis afterwards? I ask because I do not want to stretch myself too thin, because I work over forty hours per week and have a family. If it makes...
  39. G

    Real Analysis question - Show that the derivative is continuous.

    Homework Statement Suppose that f is differentiable at every point in a closed, bounded interval [a,b]. Prove that if f' is increasing on (a,b), then f' is continuous on (a,b). Homework Equations If f' is increasing on (a,b) and c belongs to (a,b), then f'(c+) and f'(c-) exist, and...
  40. J

    Withdraw from Intro. to Real Analysis or take a C?

    I was overly ambitious this semester and took on too many courses (4 math courses and 3 econ. courses). I am getting an A in all of my other courses except Intro to Real Analysis which I am doing horribly. I bombed a midterm which brought my overall grade down from an A- to a C. The only way to...
  41. B

    Very difficult Real Analysis question on Lebesgue integration

    Suppose gn are nonnegative and integrable on [0, 1], and that gn \rightarrow g almost everywhere. Further suppose that for all \epsilon > 0, \exists \delta > 0 such that for all A \subset [0, 1], we have meas(A) < \delta implies that supn \intA |gn| < \epsilon. Prove that g is integrable...
  42. Q

    Real Analysis Continuity problem.

    Homework Statement Show that |f(x) - f(y) | < |x - y| if f(x) = sqrt(4+x^2) if x is not equal to xo. What does this prove about f? Homework Equations The Attempt at a Solution Already proved the first part. I am guessing that for the second part the answer is that f is...
  43. C

    Is √(n-1) + √(n+1) Always Irrational for n≥1?

    Homework Statement Prove that √(n-1)+√(n+1) is irrational for every integer n≥1. Homework Equations Proofs i.e. by contradiction The Attempt at a Solution 2n + 2√(n^2-1) = x^2 so √(n^2-1) = (x^2-2n)/2 Now if x is rational then so is (x^2-2n)/2 so this says that √(n^2-1) is...
  44. R

    Can you think of a counter example (real analysis)

    Homework Statement X and Y are two closed non-empty subsets of R (real numbers). define X+Y to be (x+y | x belongs to X and y belongs to Y) give an example where X+Y is not closed Homework Equations The Attempt at a Solution i tried X=all integers and Y=[0 1] but that didnt work out. i know...
  45. J

    A simple Intro to Real Analysis question

    Homework Statement I'm asked to prove that If F is an ordered field, then the following properties hold for any elements a, b, and c of F: (a) a<b if and only if 0<b-a (b) ... ... Right now I'm working on (a) Homework Equations We're supposed to draw from the basic...
  46. M

    Real analysis help(countable union)

    Homework Statement Show that if E \subseteq R is open, then E can be written as an at most countable union of disjoint intervals, i.e., E=\bigcup_n(a_n,b_n). (It's possible that a_n=-\inf or b_n=+\inf for some n.) Hint: One way to do this is to put open intervals around each rational point...
  47. R

    Real Analysis 101: Tips for Writing Good Proofs

    hello everyone! I just started a course in real analysis and i must say that it is quite different from all the "engineering math" that i have taken before.I was wondering if anyone could give me tips or advice on how to get better at writing good proofs. Right now,we are using a book called...
  48. ╔(σ_σ)╝

    I am getting frustrated with this question ( Real analysis)

    Homework Statement Let A be the set of all real-valued functions on [0,1]. Show that there does not exist a function from [0,1] onto A. I spent half of my Saturday trying to prove this proposition and I couldn't make headway. Homework Equations The Attempt at a SolutionWell it only makes...
  49. E

    An intro to real analysis question. eazy?

    Homework Statement Let f : A -> B be a bijection. Show that if a function g is such that f(g(x)) = x for all x ϵ B and g(f(x)) = x for all x ϵ A, then g = f^-1. Use only the definition of a function and the definition of the inverse of a function. Homework Equations The...
  50. K

    Set of all finite subsets of N (real analysis)

    Homework Statement Show that the set of all finite subsets of N is a countable set. The Attempt at a Solution At first I thought this was really easy. I had A = {B1, B2, B3, ... }, where Bn is some finite subset of N. Since any B is finite and therefore countable, and since a union of...
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