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. M

    I Question about uniform convergence in a proof

    The below proposition is from David C. Ullrich's "Complex Made Simple" (pages 264-265) Proposition 14.5. Suppose ##D## is a bounded simply connected open set in the plane, and let ##\phi: D \rightarrow \mathbb{D}## be a conformal equivalence. (i) If ##\zeta## is a simple boundary point of...
  2. P

    Verify property of inner product

    I struggle with verifying positive-definiteness, in particular $$\langle f,f\rangle =0\implies f=0.$$ I know that for continuous non-negative functions, if the integral vanishes, then the function is identically ##0##. Here, however, ##f## being in ##L^2## does not make it continuous, right...
  3. Z

    Prove every subset of countable set is either finite or else countable

    There are a lot of steps left out of this proof. Here is my proof with hopefully all the steps. I would like to know if it is correct Let ##A## be a countable set. Then ##A## is either finite or countably infinite. Case 1: ##A## is finite. There is a bijection ##f## from ##A## onto...
  4. Z

    Prove that every subset ##B## of ##A=\{1,...,n\}## is finite

    I am very unsure about the proof below. I'd like to know if it is correct. If ##B## is empty then it is finite by definition. If ##B## is non-empty then since ##B\subset\mathbb{N}## it has a smallest element ##b_1##. If ##B \backslash \{b_1\}## is non-empty then it has a smallest element...
  5. Z

    Prove if ##x<0## and ##y<z## then ##xy>xz## (Rudin)

    These axioms lead to certain properties The properties above apply to all fields. We can define a more specific type of field, the ordered field And the following properties follow from this definition My question is about the proof of (c). My initial proof was Using b) with...
  6. Z

    Prove ##|\{ q\in\mathbb{Q}: q>0 \} |=|\mathbb{N}|##.

    This problem is the final exercise of problem set 1 on MIT OCW's course 18.100A, Real Analysis. Since there are no solutions available for this problem set, I would like to show my attempt at a solution here and ask if it is correct. Here is the problem statement (also available as problem 6...
  7. P

    I On (real) entire functions and the identity theorem

    In Ordinary Differential Equations by Adkins and Davidson, in a chapter on the Laplace transform (specifically, in a section where they discuss the linear space ##\mathcal{E}_{q(s)}## of input functions that have Laplace transforms that can be expressed as proper rational functions with a fixed...
  8. P

    B Questions on a numberphile video

    The above two video from numberphile are trying to motivate real analysis (I think?). The latter continues on from the former. Presenter's argument goes kind of like this: He first considers the real number line and talks about measuring distance between numbers 4-3=1 Then he talks about...
  9. L

    I Why is this function not ##L^1(\mathbb{R} \times \mathbb{R})##?

    Hi everyone in the following expression ##f(t)=\frac{1}{2 \pi} \int\left(\int f(u) e^{-i \omega u} d u\right) e^{i \omega t} d \omega ## the book says I can't swap integrals bacause the function ##f(u) e^{i \omega(t-u)}## is not ## L^1(\mathbb{R} \times \mathbb{R})## why ? complex...
  10. S

    On the ratio test for power series

    In these lecture notes, there is the following theorem and proof: I'm confused about "...the power series converges if ##0\leq r<1##, or ##|x-c|<R##...". In other words, why is ##|x-c|<R## equivalent to ##0\leq r<1##? I guess the author reasons as follows. If $$R=\lim _{n\to \infty...
  11. I

    Prove ##(a+b) + c = a + (b+c)## using Peano postulates

    I have to prove the associative law for addition ##(a+b) + c = a + (b+c)## using Peano postulates, given that ##a, b, c \in \mathbb{N}##. Now define the set $$ G = \{ z \in \mathbb{N} |\forall\; x, y \in \mathbb{N} \quad (x + y) + z = x + (y + z) \} $$ Obviously, ## G \subseteq \mathbb{N} ##...
  12. caffeinemachine

    I Feedback for my YouTube Videos on Real Analysis

    Some time back I posted about my videos on Group Theory on YouTube and got valuable feedback from the PF community. With the response in mind, I made substantial changes to my presentation. One of the main complaints was that I was speaking too fast. Here is my recent video on Real Analysis...
  13. bhobba

    B How can hyperreal numbers make infinitesimals logically sound in calculus?

    When I learned calculus, the intuitive idea of infinitesimal was used. These are numbers so small that, for all practical purposes (say 1/trillion to the power of a trillion) can be taken as zero but are not. That way, when defining the derivative, you do not run into 0/0, but when required...
  14. L

    Approaching the Measure of a Set: Strategies for Finding f(Eα)

    my question is how can I approch the problem ? And what is explicitly the set f(Eα)? {f(x) ∈ [a, b] such that what ??}
  15. M

    My proof of the Geometry-Real Analysis theorem

    Consider a convex shape ##S## of positive area ##A## inside the unit square. Let ##a≤1## be the supremum of all subsets of the unit square that can be obtained as disjoint union of finitely many scaled and translated copies of ##S##. Partition the square into ##n×n## smaller squares (see...
  16. J

    Proof for Real Analysis

    Proof: Suppose f is a function and x is in the domain of f s.t. there is a derivative at the point x and sppse. there are two tangent lines at the point (x,f(x)). Let t1 represent one of the tangent lines at (x,f(x)) and let t2 represent the other tangent line at (x,f(x)) s.t. the slopes of t1...
  17. C

    Problems with Real Analysis class

    Hi. I'm nearing retirement so I thought I would take some math classes. This fall I took a Real Analysis class at a good school and dropped it because I did so bad on the first exam. I did great on the homework and quizzes. I also took Real Analysis about 47 years ago at a very good school...
  18. 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...
  19. 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...
  20. 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{...
  21. H

    Proving that an Integer lies between x and y using Set Theory

    ## 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...
  22. 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...
  23. 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##...
  24. PseudoQuantum

    Analysis Which Math Textbook is Best for Preparing for Real Analysis?

    Hi all. I am a math major. I will be taking real analysis next Fall with an excellent professor who I know to be also quite demanding. I would like to be as well prepared for this class as possible besides going through a real analysis text or lecture series over the Summer and causing the class...
  25. Math Amateur

    MHB Exploring Proposition 6.1.2 from D&K's Multidimensional Real Analysis II (Integration)

    I am reading Multidimensional Real Analysis II (Integration) by J.J. Duistermaat and J.A.C. Kolk ... and am focused on Chapter 6: Integration ... I need some help with the proof of Proposition 6.1.2 ... and for this post I will focus on the first auxiliary result ... see (i) ... at the start of...
  26. 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...
  27. docnet

    Real analysis: prove the limit exists

    Prove that each of the limits exists or does not exist. 1. ##\text{lim}_{x\rightarrow 2}(x^2-1)=3## ##\text{lim}_{x\rightarrow 2}(x^2-1)=3## if ##\forall \epsilon>0, \exists \delta ## such that ##|x-2|<\delta \Rightarrow |f(x)-3|<\epsilon##. \begin{align}&|x^2-1|=|x+1||x-1|\leq \epsilon\\...
  28. 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##...
  29. 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...
  30. 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...
  31. 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...
  32. Mr.Husky

    Other Learning Real Analysis at My Own Pace

    Hi everyone, I recently started studying real analysis from baby Bruckner couple. It feels me like, "I am running too fast to reach my destination but in the process of running, I decreased my oxygen level." So, I stopped trying to complete uni coursework fast. But rather I started reading...
  33. 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...
  34. 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...
  35. 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...
  36. Falgun

    Analysis Real Analysis (Baby Rudin vs Apostol)

    I am currently trying to self study Real analysis . I have completed Hubbard's Multivariable book & Strang's Linear algebra book. I have Apostol's Mathematical Analysis & Baby Rudin . I started with rudin yesterday and was making excellent headway until I encountered a theorem about 15 pages in...
  37. 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...
  38. 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...
  39. yucheng

    Understanding the Use of Min in Cauchy Sequences

    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}...
  40. 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...
  41. 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...
  42. S

    B Is complex analysis really much easier than real analysis?

    This author seems to say so: https://blogs.scientificamerican.com/roots-of-unity/one-weird-trick-to-make-calculus-more-beautiful/
  43. 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...
  44. 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...
  45. 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...
  46. CaptainAmerica17

    I Help Finding the Correct Approach to this Proof (Intro Real Analysis)

    Ok, so here is what I have so far: Suppose ##T_1## is infinite and ##\varphi : T_1 \rightarrow T_2## is a bijection. Reasoning: I'm thinking I would then show that there is a bijection, which would be a contradiction since an infinite set couldn't possibly have a one-to-one correspondence...
  47. CaptainAmerica17

    Would someone mind checking my proof? Intro Real Analysis

    Here is my solution. I used mathjax to type it up in Overleaf. I feel like it makes sense, but I also have a feeling I might have "jumped the gun" with my logic. If it is correct, I would appreciate feedback on how to improve it. Thanks!
  48. 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...
  49. 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...
  50. 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## $$ x_{j} \leq y_n.$$...
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