Real analysis Definition and 106 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. MexChemE

    Analysis Study plan for Functional Analysis - Recommendations and critique

    Hello, PF! It’s been a while since I last posted. I am looking for a critique and recommendations regarding my study plan towards Functional Analysis and applications (convex optimization, optimal control), but first, some background: - This plan is in preparation for my master’s thesis, I...
  2. H

    Proving that there is a sequence in S, such that ##\lim s_n = \sup S##

    Let ##S=\{s_n:n∈N\}##. ##\sup S## is the least upper bound of S. For any ϵ>0, we have an m such that ##\sup S−\epsilon \lt s_m## ##\sup S−s_m \lt \varepsilon## ##|\sup S−s_m| \lt \varepsilon## I mean to say that, no matter how small ϵ is, there is always an element of S whose distance from supS...
  3. I

    If ##x> 1## and ##x^2 <2##, prove ##x < y##, ##y^2<2##

    Suppose ##x \in \mathbb{Q}## and ##x > 1## and ## x^2 < 2##. I need to come up with some ##y \in \mathbb{Q}## such that ##x < y## and ## y^2 < 2##. Here is my attempt. Give that ##x > 1## and ## x^2 < 2##, I have ## (2-x^2) > 0## and ##4x > 0##. Also, ##2x >0##. Now define $$ \alpha = \text{...
  4. H

    My proof of "if ##y-x \gt 1## then there exists an integer between them"

    ## y-x \gt 1 \implies y \gt 1+x## Consider the set ##S## which is bounded by an integer ##m##, ## S= \{x+n : n\in N and x+n \lt m\}##. Let's say ##Max {S} = x+n_0##, then we have $$ x+n_0 \leq m \leq x+(n_0 +1)$$ We have, $$ x +n_0 \leq m \leq (x+1) +n_0 \lt y+ n_0 $$ Thus, ##x+n_0 \leq m \lt...
  5. H

    Proving a property of a Dedekind cut

    A Dedekind cut is a pair ##(A,B)##, where ##A## and ##B## are both subsets of rationals. This pair has to satisfy the following properties A is nonempty B is nonempty If ##a\in A## and ##c \lt a## then ##c \in A## If ##b \in B## and ## c\gt b## then ##c \in B## If ##b \not\in B## and ## a\lt...
  6. C

    Which of the following statements are true? (Real Analysis question)

    Summary:: x Problem: Let ##f:[0, \infty) \rightarrow \mathbb{R}## be a positive function s.t. for all ## M > 0 ## it occurs that ## f ## is integrable on ## [0,M] ##. Which of the following statements are true? A. If ##\lim _{x \rightarrow+\infty} f(x)=0## then ##\int_{0}^{\infty} f(x) d x##...
  7. C

    I In Euclidian space, closed ball is equal to closure of open ball

    Problem: Let ## (X,d) ## be a metric space, denote as ## B(c,r) = \{ x \in X : d(c,x) < r \} ## the open ball at radius ## r>0 ## around ## c \in X ##, denote as ## \bar{B}(c, r) = \{ x \in X : d(c,x) \leq r \} ## the closed ball and for all ## A \subset X ## we'll denote as ## cl(A) ## the...
  8. H

    Prove that the inner product converges

    I'm learning Linear Algebra by self and I began with Apsotol's Calculus Vol 2. Things were going fine but in exercise 1.13 there appeared too many questions requiring a strong knowledge of Real Analysis. Here is one of it (question no. 14) Let ##V## be the set of all real functions ##f##...
  9. C

    I What's the definition of "periodic extension of a function"?

    I'm learning about Fourier theory from my lecture notes and I have a few questions that I wasn't able to concretely find answers to: 1. What's the definition of periodic extension? I think the definition is as follows ( Correct me if I'm wrong please ): for ## f: [ a,b) \to \mathbb{R} ## its...
  10. C

    Prove limit comparison test for Integrals

    Attempt: Note we must have that ## f>0 ## and ## g>0 ## from some place or ## f<0 ## and ## g<0 ## from some place or ## g ,f ## have the same sign in ## [ 1, +\infty) ##. Otherwise, we'd have that there are infinitely many ##x's ## where ##g,f ## differ and sign so we can chose a...
  11. U

    I Limit of limits of linear combinations of indicator functions

    I have a sequence of functions ##0\leq f_1\leq f_2\leq ... \leq f_n \leq ...##, each one defined in ##\mathbb{R}^n## with values in ##\mathbb{R}##. I have also that ##f_n\uparrow f##. Every ##f_i## is the limit (almost everywhere) of "step" functions, that is a linear combination of rectangles...
  12. Mr.Husky

    Analysis Opinions on textbooks on Analysis

    What are your opinions on Barry Simon's "A Comprehensive Course in Analysis" 5 volume set. I bought them with huge discount (paperback version). But I am not sure should I go through these books? I have 4 years and can spend 12 hours a week on them. Note- I am now studying real analysis from...
  13. C

    Showing continuous function has min or max using Cauchy limit def.

    Problem: Let ## f: \Bbb R \to \Bbb R ## be continuous. It is known that ## \lim_{x \to \infty } f(x) = \lim_{x \to -\infty } f(x) = l \in R \cup \{ \pm \infty \} ##. Prove that ## f ## gets maximum or minimum on ## \Bbb R ##. Proof: First we'll regard the case ## l = \infty ## ( the case...
  14. C

    I ##(a_n) ## has +10,-10 as partial limits. Then 0 is also a partial limit

    Problem: If sequence ## (a_n) ## has ##10-10## as partial limits and in addition ##\forall n \in \mathbb{N}.|a_{n+1} − a_{n} |≤ \frac{1}{n} ##, then 0 is a partial limit of ## (a_n) ##. Proof : Suppose that ## 0 ## isn't a partial limit of ## (a_n) ##. Then there exists ## \epsilon_0 > 0 ## and...
  15. yucheng

    Can I use recursion/induction to show that N <= x < N+1 for x real

    Homework Statement:: Show that for every real number ##x## there is exactly one integer ##N## such that ##N \leq x < N+1##. (This integer is called the integer part of ##x##, and is sometimes denoted ##N = \lfloor x\rfloor##.) Relevant Equations:: N/A I have tried reading the solution given...
  16. yucheng

    Is my proof that multiplication is well-defined for reals correct?

    I have referred to this page: https://taoanalysis.wordpress.com/2020/03/26/exercise-5-3-2/ to check my answer. The way I thought of the problem: I know ##xy = \mathrm{LIM}_{n\to\infty} a_n b_n## and I know ##x'y = \mathrm{LIM}_{n\to\infty} a'_n b_n##. Thus if ##xy=x'y##, maybe I can try showing...
  17. yucheng

    Purpose of \varepsilon' := \min\left(\frac{\varepsilon}{3M_2}, \frac\varepsilon 3\right)

    I refer to this page: https://taoanalysis.wordpress.com/2020/03/26/exercise-5-3-2/ I am having trouble understanding the purpose / motivation behind using the min as in ##\delta := \min\left(\frac{\varepsilon}{3M_1}, 1\right)## and ##\varepsilon' := \min\left(\frac{\varepsilon}{3M_2}...
  18. yucheng

    I Is ##\delta##-steady needed in this proof, given ##\epsilon##-steady

    In Tao's Analysis 1, Lemma 5.3.6, he claims that "We know that ##(a_n)_{n=1}^{\infty}## is eventually ##\delta##-steady for everyvalue of ##\delta>0##. This implies that it is not only ##\epsilon##-steady, ##\forall\epsilon>0##, but also ##\epsilon/ 2##-steady." My question is, why do we need...
  19. yucheng

    Proof that two equivalent sequences are both Cauchy sequences

    Let us just lay down some definitions. Both sequences are equivalent iff for each ##\epsilon>0## , there exists an N>0 such that for all n>N, ##|a_n-b_n|<\epsilon##. A sequence is a Cauchy sequence iff ##\forall\epsilon>0:(\exists N>0: (\forall j,k>N:|a_j-a_k|>\epsilon))##. We proceeded by...
  20. Adesh

    I Will ##M_i = m_i## if an interval is made vanishingly small?

    We define : $$M_i = sup \{f(x) : x \in [x_{i-1}, x_i ] \}$$ $$m_i = inf \{f(x) : x \in [ x_{i-1}, x_i ] \}$$ Now, if we make the length of the interval ##[x_{i-1}, x_i]## vanishingly small, then would we have ##M_i = m_i##? I have reasons for believing so because as the size of the interval is...
  21. Adesh

    I How to prove that ##f## is integrable given that ##g## is integrable?

    We have a function ##f: [a,b] \mapsto \mathbb R## (correct me if I'm wrong but the range ##\mathbb R## implies that ##f## is bounded). We have a partition ##P= \{x_0, x_1 , x_2 \cdots x_n \}## such that for any open interval ##(x_{i-1}, x_i)## we have $$ f(x) =g(x) $$ (##g:[a,b] \mapsto \mathbb...
  22. Adesh

    I Checking the integrability of a function using upper and lowers sums

    Hello and Good Afternoon! Today I need the help of respectable member of this forum on the topic of integrability. According to Mr. Michael Spivak: A function ##f## which is bounded on ##[a,b]## is integrable on ##[a,b]## if and only if $$ sup \{L (f,P) : \text{P belongs to the set of...
  23. O

    How to prove this statement about the derivative of a function

    My try: ##\begin{align} \dfrac{d {r^2}}{d r} \dfrac{\partial r}{\partial p} = \dfrac{\partial {r^2}}{\partial p} \tag1\\ \dfrac{\partial r}{\partial p} = \dfrac{\partial {r^2}}{\partial p} \dfrac{1}{\dfrac{d r^2}{d r}}=\dfrac{p-a\cos\theta}{r} \tag2\\ \end{align}## By chain rule...
  24. C

    I Finite expansion of a fraction of functions

    I am having a problem finding the right order above and below to find the finite expansion of a fraction of usual functions assembled in complicated ways. For instance, a question asked to find the limit as x approaches 0 for the following function I know that to solve it we must first find...
  25. CoffeeNerd999

    I Do I need induction to prove that this sequence is monotonic?

    I think the initial assumptions would allow me to prove this without induction. Suppose ##(x_n)## is a real sequence that is bounded above. Define $$ y_n = \sup\{x_j | j \geq n\}.$$ Let ##n \in \mathbb{N}##. Then for all ##j \in \mathbb{N}## such that ##j \geq n + 1 > n## $$ x_{j} \leq y_n.$$...
  26. D

    I Rudin: theorem 1.21

    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 ?
  27. 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...
  28. 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...
  29. 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...
  30. 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 =...
  31. U

    I Boundedness of derivatives

    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...
  32. 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
  33. NihalRi

    Proof about 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...
  34. 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...
  35. 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 ##...
  36. 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...
  37. 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...
  38. 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...
  39. 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...
  40. 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...
  41. 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...
  42. 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...
  43. 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...
  44. 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...
  45. 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...
  46. 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...
  47. A

    B A Rational Game

    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]...
  48. 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...
  49. 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...
  50. 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...
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