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. zeronem

    Introductory Real Analysis Problem

    $$r<x<s$$ $$s-r>0$$ We enploy the Archimedean principle where $$n(s-r)>1$$ We employ density of rationals where $$\exists [m,m+1] \in Q$$ Such that $$nr\in [m,m+1)$$ Therefore $$m\leq nr \lt m+1$$$$ \frac m n \leq r \lt \frac m n + \frac 1 n $$ Since $$ \frac m n \leq r $$ Then $$...
  2. AlmX

    Analysis Is Baby Rudin a good choice for first my Real Analysis textbook?

    Summary: Is Baby Rudin a good choice for first Real Analysis textbook for someone without strong pure math background? I've completed 2 semesters of college calculus, but not "pure math" calculus which is taught to math students. I'm looking for introductory text on Real Analysis and I've...
  3. D

    I Rudin Theorem 1.21: Maximizing t Value

    Summary: Rudin theorem 1.21 He has said that as t=X/(X+1) then t^n<t<1 then maximum value of t is 1. then in the next part he has given that t^n<t<x. as maximum value of t is less than 1 why has he given that t<x ?
  4. P

    Help with a real analysis problem

    I tried to prove this by absurd stating that there is no such ## \mu'## but i couldn't get anywhere...
  5. MidgetDwarf

    Intro Real Analysis: Closed and Open sets Of R. Help with Problem

    For the set A: Note that if n is odd, then ## A = \{ -1 + \frac {2} {n} : \text{n is an odd integer} \} ## . If n is even, A = ## \{1 + ~ \frac {2} {n} : \text{ n is an even integer} \} ## . By a previous exercise, we know that ## \frac {1} {n} ## -> 0. Let ## A_1 ## be the sequence when n...
  6. Tonzzi

    Real Analysis Textbook Recommendation

    Does anyone have a recommendation for a book(s) to use for the self-study of real analysis? I have just finished Apostol Calculus, Vol. 2 and would like to move on to real analysis. I am not sure whether I should continue following Apostol and move on to Apostol mathematical analysis or...
  7. V

    A What type of function satisfy a type of growth condition?

    Let ##f:\mathbb{R}^n\rightarrow\mathbb{R}^n##. Is there any class of function and some type of "growth conditions" such that bounds like below can be established: \begin{equation} ||f(x)||\geq g\left( \text{dist}(x,\mathcal{X})\right), \end{equation} with ##\mathcal{X}:= \{x:f(x)=0\}## (zero...
  8. anhtudo

    I Lemma 1.2.3 - Ethan.D.Bloch - The Real Numbers and Real Analysis

  9. S

    Positive derivative implies growing function using Bolzano-Weierstrass

    I'm stuck on a proof involving the Bolzano-Weierstrass theorem. Consider the following statement: $$f'(x)>0 \ \text{on} \ [a,b] \implies \forall x_1,x_2\in[a,b], \ f(x_1)<f(x_2) \ \text{for} \ x_1<x_2 $$ i.e. a positive derivative over an interval implies that the function is growing over the...
  10. NihalRi

    Function Continuity Proof in Real Analysis

    Homework Statement We've been given a set of hints to solve the problem below and I'm stuck on one of them Let f:[a,b]->R , prove, using the hints below, that if f is continuous and if f(a) < 0 < f(b), then there exists a c ∈ (a,b) such that f(c) = 0 Hint let set S = {x∈[a,b]:f(x)≤0} let c =...
  11. U

    I Proving Alternating Derivatives with Induction in Mathematical Analysis I

    Hi forum. I'm trying to prove a claim from Mathematical Analysis I - Zorich since some days, but I succeeded only in part. The complete claim is: $$\left\{\begin{matrix} f\in\mathcal{C}^{(n)}(-1,1) \\ \sup_{x\in (-1,1)}|f(x)|\leq 1 \\ |f'(0)|>\alpha _n \end{matrix}\right. \Rightarrow \exists...
  12. J

    MHB Real Analysis - Convergence to Essential Supremum

    Problem: Let $\left(X, M, \mu\right)$ be a probability space. Suppose $f \in L^\infty\left(\mu\right)$ and $\left| \left| f \right| \right|_\infty > 0$. Prove that $lim_{n \rightarrow \infty} \frac{\int_{X}^{}\left| f \right|^{n+1} \,d\mu}{\int_{X}^{}\left| f \right|^{n} \,d\mu} = \left| \left|...
  13. Miguel

    Single Point Continuity - Spivak Ch.6 Q5

    Hey Guys, I posed this on Math Stackexchange but no one is offering a good answering. I though you guys might be able to help :) https://math.stackexchange.com/questions/3049661/single-point-continuity-spivak-ch-6-q5
  14. NihalRi

    What is the proof for the limit superior?

    Homework Statement 2. Relevant equation Below is the definition of the limit superior The Attempt at a Solution I tried to start by considering two cases, case 1 in which the sequence does not converge and case 2 in which the sequence converges and got stuck with the second case. I know...
  15. H

    Prove that there exists a graph with these points such that....

    Homework Statement Let us have ##n \geq 3## points in a square whose side length is ##1##. Prove that there exists a graph with these points such that ##G## is connected, and $$\sum_{\{v_i,v_j\} \in E(G)}{|v_i - v_j|} \leq 10\sqrt{n}$$ Prove also the ##10## in the inequality can't be replaced...
  16. F

    Curve and admissible change of variable

    Homework Statement If I have the two curves ##\phi (t) = ( \cos t , \sin t ) ## with ## t \in [0, 2\pi]## ##\psi(s) = ( \sin 2s , \cos 2s ) ## with ## s \in [\frac{\pi}{4} , \frac{5 \pi}{4} ] ## My textbook says that they are equivalent because ##\psi(s) = \phi \circ g^{-1}(s) ## where ##...
  17. T

    Need help formalizing "T is an open set"

    Homework Statement Let ##S\subseteq \Bbb{R}## and ##T = \{ t\in \Bbb{R} : \exists s\in S, \vert t-s\vert \lt \epsilon\}## where ##\epsilon## is fixed. I need to show T is an open set. Homework Equations n/a The Attempt at a Solution Let ##x \in T##, then ##\exists \sigma \in S## such that ##x...
  18. T

    Image of a f with a local minima at all points is countable.

    Homework Statement Let ##f:\Bbb{R} \to \Bbb{R}## be a function such that ##f## has a local minimum for all ##x \in \Bbb{R}## (This means that for each ##x \in \Bbb{R}## there is an ##\epsilon \gt 0## where if ##\vert x-t\vert \lt \epsilon## then ##f(x) \leq f(t)##.). Then the image of ##f## is...
  19. A

    I Learning the theory of the n-dimensional Riemann integral

    I would like to learn (self-study) the theory behind the n-dimensional Riemann integral (multiple Riemann integrals, not Lebesgue integral). I am from Croatia and found lecture notes which Croatian students use but they are not suitable for self-study. The notes seem to be based on the book: J...
  20. J

    MHB Real Analysis, Sequences in relation to Geometric Series and their sums

    I will state the problem below. I don't quite understand what I am needing to show. Could someone point me in the right direction? I would greatly appreciate it. Problem: Let p be a natural number greater than 1, and x a real number, 0<x<1. Show that there is a sequence $(a_n)$ of integers...
  21. J

    MHB Real Analysis, liminf/limsup Equality and Multiplication

    Here are a couple more problems I am working on! Problem 1: Prove that, $limsupa_n+liminfb_n \leq limsup(a_n+b_n) \leq limsupa_n+limsupb_n$ Provided that the right and the left sides are not of the form $\infty - \infty$. Proof: Consider $(a_n)$ and $(b_n)$, sequences of real numbers...
  22. J

    MHB Real Analysis, liminf/limsup inequality

    I am working a bunch of problems for my Real Analysis course.. so I am sure there are more to come. I feel like I may have made this proof too complicated. Is it correct? And if so, is there a simpler method? Problem: Show that $liminfa_n \leq limsupa_n$. Proof: Consider a sequence of real...
  23. M

    I Two questions about derivatives

    In Rudin, the derivative of a function ##f: [a,b] \to \mathbb{R}## is defined as: Let ##f## be defined (and real-valued) on ##[a,b]##. For any ##x \in [a,b]##, form the quotient ##\phi(t) = \frac{f(t) - f(x)}{t-x}\quad (a < t <b, t \neq x)## and define ##f'(x) = \lim_{t \to x} \phi(t)##, if the...
  24. M

    I Question regarding a sequence proof from a book

    I have a Dover edition of Louis Brand's Advanced Calculus: An Introduction to Classical Analysis. I really like this book, but find his proof of limit laws for sequences questionable. He first proves the sum of null sequences is null and that the product of a bounded sequence with a null...
  25. T

    Show that ##\frac{1}{x^2}## is not uniformly continuous on (0,∞).

    Homework Statement Show that ##f(x)=\frac{1}{x^2}## is not uniformly continuous at ##(0,\infty)##. Homework Equations N/A The Attempt at a Solution Given ##\epsilon=1##. We want to show that we can compute for ##x## and ##y## such that ##\vert x-y\vert\lt\delta## and at the same time ##\vert...
  26. T

    Distance of a point from a compact set in ##\Bbb{R}##

    Homework Statement Let ##K\neq\emptyset## be a compact set in ##\Bbb{R}## and let ##c\in\Bbb{R}##. Then ##\exists a\in K## such that ##\vert c-a\vert=\inf\{\vert c-x\vert : x\in K\}##. 2. Relevant results Any set ##K## is compact in ##\Bbb{R}## if and only if every sequence in ##K## has a...
  27. T

    Showing that an exponentiation is continuous -- Help please....

    Homework Statement Let ##p\in\Bbb{R}##. Then the function ##f:(0,\infty)\rightarrow \Bbb{R}## defined by ##f(x):=x^p##. Then ##f## is continuous. I need someone to check what I've done so far and I really need help finishing the last part. I am clueless as to how to show continuity for...
  28. S

    A Derivation of a complex integral with real part

    Hey, I tried to construct the derivation of the integral C with respect to Y: $$ \frac{\partial C}{\partial Y} = ? $$ $$ C = \frac{2}{\pi} \int_0^{\infty} Re(d(\alpha) \frac{exp(-i \cdot ln(f))}{i \alpha}) d \alpha $$ with $$d(\alpha) = exp(i \alpha (b + ln(Y)) - u) \cdot exp(v(\alpha) + z...
  29. T

    Regarding Real numbers as limits of Cauchy sequences

    Homework Statement Let ##x\in\Bbb{R}## such that ##x\neq 0##. Then ##x=LIM_{n\rightarrow\infty}a_n## for some Cauchy sequence ##(a_n)_{n=1}^{\infty}## which is bounded away from zero. 2. Relevant definitions and propositions: 3. The attempt at a proof: Proof:(by construction) Let...
  30. Eclair_de_XII

    If A is dense in [0,1] and f(x) = 0, x in A, prove ∫fdx = 0.

    Homework Statement "A set ##A\subset [0,1]## is dense in ##[0,1]## iff every open interval that intersects ##[0,1]## contains ##x\in A##. Suppose ##f:[0,1]\rightarrow ℝ## is integrable and ##f(x) = 0,x\in A## with ##A## dense in ##[0,1]##. Show that ##\int_{0}^{1}f(x)dx=0##." Homework...
  31. Mr Davis 97

    Real Analysis Definition and Explanation

    Homework Statement 1) Suppose ##t## is a subsequential limit for ##(s_n)##. Write the precise definition of the meaning of this statement. 2) Explain why there exists a strictly increasing sequence ##(n_k)^\infty_{k=1}## of natural numbers such that ##\lim s_{n_k}=t##. Homework EquationsThe...
  32. A

    B A Rational Game: Exploring the Paradox of Aligning Irrational Numbers

    This post is to set forth a little game that attempts to demonstrate something that I find to be intriguing about the real numbers. The game is one that takes place in a theoretical sense only. It starts by assuming we have two pieces of paper. On each is a line segment of length two: [0,2]...
  33. Mr Davis 97

    Real Analysis: Prove Upper Bound of Sum of Bounded Sequences

    Homework Statement Suppose that ##( s_n )## and ## (t_n)## are bounded sequences. Given that ##A_k## is an upper bound for ##\{s_n : n \ge k \}## and ##B_k## is an upper bound for ##\{t_n : n \ge k \}## and that ##A_k + B_k## is an upper bound for ##\{s_n + t_n : n \ge k \}##, show that ##\sup...
  34. anon3335

    Rudin POMA: chapter 4 problem 14

    Homework Statement Question: Let ##I = [0,1]##. Suppose ##f## is a continuous mapping of ##I## into ##I##. Prove that ##f(x) = x## for at least one ##x∈I##. Homework Equations Define first(##[A,B]##) = ##A## and second(##[A,B]##) = ##B## where ##[A,B]## is an interval in ##R##. The Attempt at...
  35. C

    MHB Prove this proposition 2.1.13 in Induction to Real Analysis by Jiri Lebel

    Dear Everybody, I need some help with seeing if there any logical leaps or any errors in this proves. Corollary 1.2.8 to Proposition 1.2.8 states: if $S\subset\Bbb{R}$ is a non-empty set, bounded from below, then for every $\varepsilon>0$ there exists a $y\in S$ such that $\inf...
  36. C

    MHB Prove this thereom 1.2.6 v in Introduction to Real Analysis by Jiri Lebl

    Prove this Proposition 1.2.6 v in Introduction to Real Analysis by Jiri Lebl Dear Everybody, I need some help with seeing if there are any logical leaps or errors in this proof. The theorem states: $A\subset\Bbb{R}$ and $A\ne\emptyset$ If $x<0$ and A is bounded below, then...
  37. Math Amateur

    MHB Multidimensional Real Analysis - Duistermaat and Kolk, Lemma 1.1.7 ....

    I am reading "Multidimensional Real Analysis I: Differentiation by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of Lemma 1,1,7 (ii) ... Duistermaat and Kolk"s Lemma 1.1.7 reads as follows: In the above Lemma part (ii)...
  38. S

    Convergence of a double summation using diagonals

    Homework Statement Show that ##\sum_{k=2}^\infty d_k## converges to ##\lim_{n\to\infty} s_{nn}##. Homework Equations I've included some relevant information below: The Attempt at a Solution So far I've managed to show that ##\sum_{k=2}^\infty |d_k|## converges, but I don't know how to move...
  39. T

    Intro to analysis, intro to real analysis I, numerical analysis

    Hello, Is there a difference from these courses, or are they the same course with different names? I need to know which one to choose for the upcoming semester... Intro to Analysis, Intro to Real Analysis I, and Numerical Analysis Thank you, Tracie
  40. S

    Generalization of a theorem in Real Analysis

    Homework Statement If ##\{K_\alpha\}## is a collection of compact subsets of a metric space X, such that the intersection of every finite subcollection of {##K_\alpha##} is nonempty, then ##\cap K_\alpha## is nonempty. Generalize this theorem and proof the generalization. Why doesn't it make...
  41. R

    A Solution of a weakly formulated pde involving p-Laplacian

    Let $$f:\Omega\to\mathbb{R}$$, where $$\Omega\subset\mathbb{R}^d$$, and $$\Omega$$ is convex and bounded. Let $$\{x_i\}_{i=1,2,..N}$$ be a set of points in the interior of $$\Omega$$. $$d_i\in\mathbb{R}$,$i = 1,2,..N$$ I want to solve this weakly formulated pde: $$ 0=\frac{A}{N^{d+1}} \sum_i...
  42. D

    Other Books for Geometry, Real Analysis and EM

    Hi, all. I would like to read books about the topics - Geometry, Real Analysis and Electricity and Magnetism. And I find the followings. Are they decent and rigorous? Geometry The Real Numbers and Real Analysis Introduction to Electrodynamics Classical Electricity and Magnetism Electricity...
  43. O

    Schools Math Grad school with only one real analysis course?

    Assume student has taken around 8 upper division math courses including abstract algebra 1 and abstract algebra 2.
  44. DavideGenoa

    I Differentiation under the integral in retarded potentials

    Hello, friends! I know, thanks to @Hawkeye18 who proved this identity to me, that, if ##\phi:V\to\mathbb{R}## is a bounded measurable function defined on the bounded measurable domain ##V\subset\mathbb{R}^3##, then, for any ##k\in\{1,2,3\}##, $$\frac{\partial}{\partial r_k}\int_V...
  45. DavideGenoa

    I Laplacian of retarded potential

    Dear friends, I have found a derivation of the fact that, under the assumptions made in physics on ##\rho## (to which we can give the physical interpretation of charge density) the function defined by $$V(\mathbf{x},t):=\frac{1}{4\pi\varepsilon_0}\int_{\mathbb{R}^3}...
  46. Anshul23

    Light in a cup (Can you explain this phenomenon?)

    Can anyone explain the behavior of light I came across as I sat in my lounge this evening having a nice cup of Mocha . Hint ( I am sitting in a room with some led ceiling lights on) can you: 1.Guess how many Led lights I have on 2.Explain the appearance of light which is looking like a typical...
  47. Derek Hart

    Spivak Chapter 5 Problem 26) a

    Homework Statement Give an example to show that the given "definition" of limx→aƒ(x) = L is incorrect. Definition: For each 0<δ there is an 0<ε such that if 0< l x-a I < δ , then I ƒ(x) - L I < ε . Homework EquationsThe Attempt at a Solution I considered the piece-wise function: ƒ(x) = (0 if...
  48. S

    B Real Analysis: Expanding a Function at Different Points

    Hello! Can someone explain to me, in real analysis, what is the difference in expanding a function as a Taylor series around 2 different point. So we have ##f(x)=\sum c_k (z-z_1)^k = \sum d_k (z-z_2)^k## and as ##k \to \infty## the series equals f in both cases, but why would one choose a point...
  49. S

    I Difference between complex and real analysis

    Hello! I see that all theorems in complex analysis are talking about a function in a region of the complex plane. A region is defined as an open, connected set. If I am not wrong, the real line, based on this definition, is a region. I am a bit confused why there are so many properties of the...
  50. G

    Courses Which version of Real Analysis to take?

    Hello,I am a mechanical engineering student that loves mathematics and fluid mechanics. My school offers three different analysis courses and I’m not sure which to take. I took honors Fundamental of Mathematics, where we covered Abstract Linear Algebra, Set theory (along with rings and fields)...
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