Real analysis Definition and 524 Threads

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...

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)...

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...

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

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...

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...

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...

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

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...

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}...

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...

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...

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...

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...

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)...

Claim: Let A be a non-empty subset of R+ = {x ∈ R : x > 0} which is bounded above, and let B = {x2 : x ∈ A}. Then sup(B) = sup(A)2. a. Prove the claim. b. Does the claim still hold if we replace R+ with R? Explain briefly. So I have spent the past hours trying to prove this claim using the...

Just a couple questions. Problem 2: Just would like to know if this is the correct approach for this problem. Problem 3: I am just wondering if I can use Problem 2 to prove the first part of Problem 3? Because to me, they seem very similar. Problem 4: Would I use the MVT for integrals...

I have no idea how to incorporate the limit into the basic definitions for a Riemann integral? All we have learned so far is how to define a Riemann integral and the properties of Riemann integrals. What should I be using for this?

Homework Statement Let (a_n) be a bounded sequence. Prove that the set of subsequential limit points of (a_n) is a subsequentially compact set Homework Equations To be a subsequentutially compact set, every sequence in the set of limit points of (a_n) must have a convergent subsequence. The...

Dear all, I currently a student in mechanical engineering and i reached the conclusion that maths from the point of view of mathematicians is lot more interesting than the eyes of engineers (for me at least). One of my friends in the maths department suggested to me to read real...

Homework Statement show that \lim_{n->\infty} \frac{n^2}{2^n} = 0 Homework Equations squeeze theorem The Attempt at a Solution I tried to use squeez theorem. I don't know how to do it because don't know how to reduce 2^n However, I can solve this question like this. Given \epsilon>0...

From Courant's Differential and Integral Calculus p.13, In an ordinary system of rectangular co-ordinates, the points for which both co-ordinates are integers are called lattice points. Prove that a triangle whose vertices are lattice points cannot be equilateral. Proof: Let ##A=(0,0)...

Suppose I wanted to prove the work-kinetic energy theorem. This means that I want to show that \frac{1}{2}m( \vec {v}^2_f - \vec{v}^2_i)=\int_{x_1}^{x_2} \vec{F} \cdot dx. So, I go ahead and start on the right side: \int_{x_1}^{x_2} (m \frac{d\vec{v}}{dt}) \cdot dx = m \int_{x_1}^{x_2}...

Homework Statement Find the range ##y = \sqrt{\ln({\cos(\sin (x)}))}## Homework EquationsThe Attempt at a Solution [/B] https://www.desmos.com/calculator I used a graphing calculator to find the intersection between ##y = e^{x^2}## and ##y = \cos(\sin(x))##. Which I get as ##(0,1)##. So the...

Hi all, I am currently in my first semester of my sophomore year, taking Real Analysis I. This class covers formal proofs, properties of the real line, sequences, series, limits, continuity and differentiation, and Riemann Integration. I apparently got stuck with the worst professor at my...

Homework Statement Given ##v = {\left\{ {v}_{i} \right\}}_{i = 1}^{\infty}## and defining ## {v}_{n}^{\left( k \right)} = {v}_{n - k} ## (Shifting Operator). Prove that there exist ## \alpha > 0 ## such that $$ \sum_{k = - \infty}^{\infty} {2}^{- \left| k \right|} \left \langle {v}^{\left (...

The definition of these relations as given in my textbook are : (1):- Reflexive :- A relation ##R : A \to A## is called reflexive if ##(a, a) \in R, \color{red}{\forall} a \in A## (2):- Symmetric :- A relation ##R : A \to A## is called symmetric if ##(a_1, a_2) \in R \implies (a_2, a_1) \in R...

Given :- $$g(f(x_1)) = g(f(x_2)) \implies x_1 = x_2$$ Question :- Check whether ##g(x)## is injective or not. Now this is of-course false; counter examples are easy to provide. But I proved that ##g(x)## must be one-one even after knowing the fact it must not. Here is the proof :- Let...

Hello, I am taking a class in RA, where we're using Bartle/Sherbert. Since I have studied few chapters from it in the summer before, I decided to take a look at a more rigorous book, like baby rudin, but since many have advised against that book, I turned to Pugh's real mathematical analysis...

I've been out of school for a while and working as a programmer. I want to start taking some masters courses for applied math (PDEs, numerical analysis, etc) and need to become familiar again with the advanced math I used to use in undergrad. I took two semesters of real analysis as an...

Suppose that we have a 2\pi-periodic, integrable function f: \mathbb{R} \rightarrow \mathbb{R}, whose continuous Fourier coefficients \hat{f} are known. The convolution theorem tells us that: $$\displaystyle \widehat{{f^2}} = \widehat{f \cdot f} = \hat{f} \ast \hat{f},$$ where \ast denotes the...

So I decide to self-study the real analysis (measure theory, Banach space, etc.). Surprisingly, I found that Rudin-RCA is quite readable; it is less terse than his PMA. Although the required text for my introductory analysis course was PMA, I mostly studied from Hairer/Wanner's Analysis by Its...

Hi, friends! I read that, if ##f\in L^1[c,d]## is a Lebesgue summable function on ##[a,b]## and ##g:[a,b]\to[c,d]## is a differomorphism (would it be enough for ##g## to be invertible and such that ##g\in C^1[a,b]## and ##g^{-1}\in C^1[a,b]##, then...

A function f: R->R is a continuous function such that f(q) = q for every rational number q. Prove f(y) = y for every real number y. I know every irrational number is the limit to a sequence of rational numbers. But I not sure how to prove f(y) = y for every real number y. Any ideas?

The most common proof that I have found of the fact that Ampère's law is entailed by the Biot-Savart law essentially uses the fact that, if ##\boldsymbol{J}:\mathbb{R}^3\to\mathbb{R}^3##, ##\boldsymbol{J}\in C_c^2(\mathbb{R}^3)##, is a compactly supported twice continuously differentiable field...

Homework Statement Prove $$T\int_c^d f(x,y)dy = \int_{c}^dTf(x,y)dy$$ where $$T:\mathcal{C}[a,b] \to \mathcal{C}[a,b]$$ is linear and continuous in L^1 norm on the set of continuous functions on [a,b] and $$f:[a,b]\times [c,d]$$ is continuous. Homework EquationsThe Attempt at a Solution [/B]...

I'm currently a first year MathPhys student, and next year I have to decide my stream. I can pick a combination (pure) Mathematics, Applied & Computationtal. Mathematics, Statistics, MathSci, Physics, Theoretical Physics or Physics with Astronomy & Space. Naturally there are restrictions, and I...

Let S and T be subsets of R such that s < t for each s ∈ S and each t ∈ T. Prove carefully that sup S ≤ inf T. Attempt: I start by using the definition for supremum and infinum, and let sup(S)= a and inf(T)= b i know that a> s and b< t for all s and t. How do i continue? , do i prove it...

Every math major eventually learns logic and standard proof techniques. For example, to show that a rigorous statement P implies statement Q, we suppose the statement P is true and use that to show Q is true. This, along with the other general proof techniques are very broad. A math major would...

Hi, I can derive a few properties of the limit inferior and limit superior of a sequence of sets but I have trouble in understanding what they actually mean. However, my understand of lim inf and lim sup of a sequence isn't all that bad. Is there a way to understand them intuitively (something...

Dear Physics Forum personnel, I recently got interested in the art of abstract proof, where the focus is writing the proof as general as possible rather than starting with a specific cases. Could anyone recommend an analysis book at the level of Rudin's PMA that treats the introductory...

Hello, Please take a look at this handbook of derivatives and integrals: http://myhandbook.info/form_diff.html http://integral-table.com/downloads/single-page-integral-table.pdf I would appreciate it if someone could point me in the direction of exemplary books that derive these...

So the course I'm taking doesn't have a textbook requirement just lecture notes as the study material. While these are sufficient I would like to supplement with an outside reference that is a bit more in depth / explanatory. It's your typical undergrad real analysis course covering: The least...

Dear Physics Forum personnel, I am a undergraduate student with math and CS major who is currently taking an introductory analysis course called MATH 521 (Rudin-PMA). On the next semester, I will be taking the course called MATH 522, which is a sequel to 521. My impression is that 522 will be...

Dear Physics Forum personnel, I am a college student with huge enthusiasm to the analysis and theoretical computer science. In order to start my journey to the real analysis. I am currently taking an introductory-analysis course (Rudin-PMA; I also use Shilov too) and linear algebra...

Homework Statement Suppose E1 and E2 are a pair of compact sets in Rd with E1 ⊆ E2, and let a = m(E1) and b=m(E2). Prove that for any c with a<c<b, there is a compact set E withE1 ⊆E⊆E2 and m(E) = c. Homework Equations m(E) is ofcourese referring to the outer measure of E The Attempt at a...

Hi I am trying to teach myself Measure Theory and I am using the book: Real Analysis by Stein and Skakarchi from Princeton. I want to check if my answers to the questions are correct, so I am asking: Does anyone have the answers to the questions in chapter 1 ?

Hi, I am trying to prove that every convergent sequence is Cauchy - just wanted to see if my reasoning is valid and that the proof is correct. Thanks! 1. Homework Statement Prove that every convergent sequence is Cauchy Homework Equations / Theorems[/B] Theorem 1: Every convergent set is...

Hi guys, my first question is:what i really need to understand real analysis? and the second is on the title:could some of you recommend a good book on real analysis? cause I've found some texts that are very difficult to understand some concepts...

Hi All, A simple question but one for which I cannot seem to get the intuition. 1. Homework Statement Find the interior point of {1/n : n ∈ ℕ}. Homework Equations N/A The Attempt at a Solution Let S = {1/n : n ∈ ℕ}, where S ⊆ℝ x is an interior point if ∃N(x ; ε), N(x ; ε) ⊆ S. My...