Real analysis Definition and 509 Threads

Homework Statement : [/B]Prove that if xn is a sequence such that |xn - xn+1| ≤ (1/3n), for all n = 1,2,..., then it converges.Homework Equations : [/B]The definition of convergence.The Attempt at a Solution :[/B] I attempted to prove this by induction, so I am clearly far off the mark here...

Homework Statement : [/B]Let a = sup S. Show that there is a sequence x1, x2, ... ∈ S such that xn converges to a.Homework Equations : [/B]I know the definition of a supremum and convergence but how do I utilize these together?The Attempt at a Solution :[/B] Given a = sup S. We know that a =...

I forgot to formally introduce myself on this forum. I am VKnopp. I am 14 year old maths enthusiast with Asperger's Syndrome. I self-educated myself all the way up to Calculus III with a little bit of number theory, linear algebra, complex analysis and real analysis supplements. I am in the top...

Let ##\mathbb{X}## be the set of all sequences in ##\mathbb{R}## that converge to ##0##. For any sequences ##\{x_n\},\{y_n\}\in\mathbb{X}##, define the metric ##d(\{x_n\},\{y_n\})=\sup_{n}{|x_n−y_n|}##. Show the metric space ##(\mathbb{X},d)## is separable. I understand that I perhaps need to...

Econ Major here. I plan to graduate in spring 2016 and from there apply to economics grad programs. I still need to take Advanced Math and Advanced Calculus, and Real Analysis, all of which are not available during the summer at my uni (Florida International University). Anyone know of any...

What are some rigorous theoretical books on mathematics for each branch of it? I have devised a fantastic list of my own and would like to hear your sentiments too. Elementary Algebra: Gelfand's Algebra Gelfand's Functions & Graphs Burnside's Theory of Equations Euler's Analysis of the...

Dear Physics Forum advisers, I am a college sophomore in US with a major in mathematics, and an aspiring algebraic number theorist and cryptographer. I wrote this email to seek your advice about taking the Analysis I (Real Analysis I), Abstract Algebra I, and Linear Algebra with Proofs. At...

Hey everyone, I'm transferring into UIUC this fall, and I just registered for my classes earlier today. I'm completing dual degrees in physics and math. I've completed the introductory physics sequence, and the introductory calculus sequence, plus a 200 level introductory differential equations...

I'm interested in watching videos of Real Analysis lectures etc. in good quality resolution. Those Harvey Mudd College lectures are valuable but annoying re video quality. Thanks. - Blue

Homework Statement Hi Guys, This is the first exampe from Engel's problem solving book. After a long period of no math I am self studying. I do not know where my knowledge deficits lie, and was recommended this site for help. "E1. Starting with a point S (a, b) of the plane with 0 < b < a...

I am working on a problem##^{(1)}## in Measure & Integration (chapter on Product Measures) like this: Suppose that ##f## is real-valued and integrable with respect to 2-dimensional Lebesgue measure on ##[0, 1]^2## and also ##\int_{0}^{a} \int_{0}^{b} f(x, y) dy dx = 0## for all ##a, b \in...

On page 671 Mary Boas has her Theorem III for that chapter. Roughly it tells us that if f(z) -a complex function- is analytic in a region, inside that region f(z) has derivatives of all orders. We can also expand this function in a taylor series. I get the part about a Taylor series, that's...

Homework Statement I came across a problem where f: (-π/2, π/2)→ℝ where f(x) = \sum\limits_{n=1}^\infty\frac{(sin(x))^n}{\sqrt(n)} The problem had three parts. The first was to prove the series was convergent ∀ x ∈ (-π/2, π/2) The second was to prove that the function f(x) was continuous...

Homework Statement Homework EquationsThe Attempt at a Solution I have no idea on how to start proving this, but I know the theorem is stating that the integral of a translated periodic function is the same with the integral of the periodic function without translation, is this concept...

I am think what is the structure of generated ##\sigma##-algebra. Let me make it specific. How to represent ##\sigma(\mathscr{A})##, where ##\mathscr{A}## is an algebra. Can I use the elements of ##\mathscr{A}## to represent the element in ##\sigma(\mathscr{A})##?

Suppose ##\mu:\mathcal{F}\rightarrow[0,\infty)## be a countable additive measure on a ##\sigma##-algebra ##\mathcal{F}## over a set ##\Omega##. Take any ##E\subseteq \Omega##. Let ##\mathcal{F}_{E}:=\sigma(\mathcal{F}\cup\{E\})##. Then, PROVE there is a countable additive measure...

Homework Statement I want to prove that the span of $\{x^{2n}:n \geq0\}$ is dense in $C([0,1])$. Furthermore, that the closure of the span of $\{x^{2n+1}:n \geq0\}$ is $\{f \in C([0,1]):f(0) = 0\}$.Homework Equations Is my solution correct? Now I do not know how to tackle the second part...

Dear Physics Forum friends, I am a sophomore in US with double majors in mathematics and microbiology. I am interested in self-studying the real analysis starting now since it will help me on my current research on computational microbiology, prepare for upcoming math research (starting on...

Homework Statement \ell is the set of sequences of real numbers where only a finite number of terms is non-zero, and the distance metric is d(x,y) = sup|x_n - y_n|, for all n in naural-numbers then the sequence u_k = {1,\frac{1}{2},\frac{1}{3},...,\frac{1}{k}, 0,0,0...} and...

Hello, Solving last exam and stuck in this exercise Homework Statement Consider an increasing sequence {xn} . We suppose ∃ x∈ℝ and {xnk} a sebsequence of {xn} and xnk→x a/ Show that for any n∈ℕ , ∃ k∈ℕ as n≤nk b/ Show that xn→x Homework Equations 3. The Attempt at a Solution [/B] For b/ it...

Hello, I have some trouble to solve this exercise Homework Statement E={ (-1)n (8n+7)/(4n-1) : n ∈ℕ} Show that 2∈[PLAIN]http://www.ilemaths.net/img/smb-bleu/derivepartielle.gifE Homework EquationsThe Attempt at a Solution We have to show that (2-r,2+r)∩ E ≠∅ and (2-r,2+r)∩ ℝ/E ≠∅ If I take...

Let A(a, b, c) and A'(a′,b′,c′) be two distinct points in R3. Let f from [0 , 1] to R3 be defined by f(t) = (1 -t) A + t A'. Instead of calling the component functions of f ,(f1, f2, f3) let us simply write f = (x, y, z). Express x; y; z in terms of the coordinates of A and A, and t. I thought...

Let U={(x,y) in R2:x2+y2<4}, and let f(x,y)=√.(4−x2−y2) Prove that f is differentiable, and find its derivative. I do know how to prove it is differentiable at a specific point in R2, but I could not generalize it to prove it differentiable on R2. Any hint?

I have this question on outer measure from Richard Bass' book, supposed to be an introductory but I am lost: Prove that ##\mu^*## is an outer measure, given a measure space ##(X, \mathcal A, \mu)## and define ##\mu^*(A) = \inf \{\mu(B) \mid A \subset B, B \in \mathcal A\}## for all subsets...

Homework Statement Let f: R -> R be a function such that \lim_{z\to 0^+} zf(z) \gt 0 Prove that there is no function g(x) such that g'(x) = f(x) for all x in R. Homework Equations Supposed to use the mean value theorem. If f(x) is continuous on [a,b] and differentiable on (a,b) then...

Homework Statement $$I = \int_{-\infty}^{\infty} e^{-x^2} dx$$ Homework Equations Below The Attempt at a Solution $$I = \int_{-\infty}^{\infty} e^{-x^2} dx$$ I don't understand, we say: $$I = \int_{-\infty}^{\infty} e^{-x^2} dx$$ Then we say: $$I = \int_{-\infty}^{\infty} e^{-t^2} dt$$...

Hi, I was leafing through some old exams of our Real analysis course, and I found this puzzling problem: "Let A⊂ℝ be Lebesgue-measurable so that for all a∈A, i = 1,2, ... (1) m1( {x∈ℝ | a+(3/4)i-2 < x < a + i-2} ) < i-3 Claim: m1(A) = 0." Initially I thought this may have something to do...

Homework Statement Please refer to : http://math.stackexchange.com/questions/1068948/how-to-prove-that-int-0-infty-sinx-arctan-frac1x-mathrm-dx-fra/1069065#1069065 The answer by @venus. What is the procedure in converting that single integral, dividing it into parts, and making it a double...

Homework Statement Establish the Inequality ##f^*(x)\ge \frac{c}{|x|ln\frac{1}{x}}## for ##f(x)=\frac{1}{|x|(ln\frac{1}{x})^2}## if ##|x|\le 1/2## and 0 otherwise Homework Equations ##f^*(x)=\sup_{x\in B} \frac{1}{m(B)} \int_B|f(y)|dy \quad x\in \mathbb{R}^d## The Attempt at a Solution...

Homework Statement Let a1=0, a2=1, and a(n+2)=n*a(n+1)+an/n+1 a)Calculate the value of a6 and a7 b)Prove that (an) converges. c)Show that lim an=1-e-1 when n goes to infinity.The Attempt at a Solution I got the a part and found out that a6 19/30 and a7)91/144 part b) each subsequent term...

Hello. The university I attend allows you to challenge some courses for grad school. Two of them are Real Analysis I and Real Analysis II. This made me consider trying it. The cost of the challenge test is the same as the cost if you took the course so it really has to count. I wanted to...

Homework Statement r = 2\cos(\theta) Homework EquationsThe Attempt at a Solution Hello, please do not evaluate. Why do textbook state that the derivative of the polar function (symbolic) is dy/dx and not dr/d\theta? It is a function of theta, then why is the derivative dy/dx? Idea: Even...

Homework Statement Show that ##\lim_{x \to a} f(x) = L## if and only if ##\lim_{x \to 0} f(x+a) = L## Homework Equations - The Attempt at a Solution For the forward direction (ie ##1 \Rightarrow 2##), I tried to first assume that 1. holds true (ie ##\forall \epsilon>0, \exists \delta>0...

Task in real analysis: P is a uniform partition on [0, π] and is divided into 6 equal subintervals. Show that the lower and upper riemann sums of sin (x) over P is lesser than 1.5 and larger than 2.4 respectively. My attempt at the solution: The greates value and the least value of sin x over...

While attempting Rudin's Principles of Mathematical Analysis, I only got about as far as page 9 before losing him in the proof that ##\mathbb{Q}## is dense in ##\mathbb{R}##. While his proof is only a few lines long, it does reveal some important properties that result from this theorem...

I am in my first semester of university and currently taking Linear Algebra. I was planning on majoring in EECS but I lost interest in EE and engineering in general (except software) and gained a lot of interest in maths (especially statistics and financial mathematics) so I will double major in...

Homework Statement Let ##A\subset E^n## and let ##f:A\to E^m.## Consider the condition that there exist some ##M\in\mathbb{R}## such that ##d(f(x),f(y))\le Md(x,y)## for all ##x,y\in A.## Show that if the condition is satisfied, if ##m=n##, and ##\text{vol}(A)=0##, then...

ignore Sorry for the post. I'll take it down soon. Thanks for the help

I'd like to start learning at home real analysis. Now, in order to start there was an older book Techniques of mathematical analysis by Tanter which looks good as preparation. I also saw that another user SanjeevGupta studied the same one, and found it good. I'd like to see some comments on...

I'm about to start scheduling my courses for next year, and I have the option of taking either Real Analysis or Complex Analysis. I'm double majoring in Math and Physics, and I want to go to grad school to study either Applied Mathematics or Physics. I haven't taken any higher level math...

Homework Statement "Formulate and prove an inequality which follows from Taylor's theorem and which remains valid for vector-valued functions." Homework Equations I know that Taylor's theorem generally states that if f is a real function on [a,b], n is a positive integer, f^{(n-1)} is...

Homework Statement let bk>0 be real numbers such that Ʃ bk diverges. Show that the series Ʃ bk/(1+bk) diverges as well. both series start at k=1Homework Equations From the Given statements, we know 1+bk>1 and 0<bk/(1+bk)<1 The Attempt at a Solution I've tried using comparison test but cannot...

Homework Statement Let f be a real function defined on the interval [a,b]/0<a<b:\forall x,y\in[a,b],x\neq y/|f(x)-f(y)|<k|x^{3}-y^{3}| where k is a positive real constant. Homework Equations 1- Prove that f is uniformly continuous on [a,b] 2- We define a function g on [a,b] such that...

Homework Statement Let f : (a, b) → R be a continuous function on (a, b) such that |f'(x)| <= 1 for all x that are elements of (a,b). Prove that f is uniformly continuous function on (a,b). Homework Equations The Attempt at a Solution Proof:For the sequence {xn}, where the limit(n→∞) xn = xo...

1) For the following choice of A, construct a function f: R → R that has discontinuities at every point x in A and is continuous on the complement of A. A = { x : 0 < x < 1} My function is f(x) = 10 if x in (0,1) and Q and f(x) = 20, if x in (0,1) and irrational number, f(x) = 30, elsewhere...

Real Analysis, L∞(E) Norm as the limit of a sequence. || f ||_{\infty} is the lesser real number M such that | \{ x \in E / |f(x)| > M \} | = 0 ( | \cdot | used with sets is the Lebesgue measure). Definition: For every 1 \leq p < \infty and for every E such that 0 < | E | < \infty we...

For the following example:(if possible give example or just state impossible 1) a bounded subset A of R for which sup A is not a limit point of A. An example is (0,1) union {7}. will this work? 2) a finite subset A of R that is not closed I think it is not possible. Please give some hints...

Homework Statement Show that the sequence of functions ##x(1-x), x^2(1-x),...## converges uniformly on ##[0,1].## 2. The attempt at a solution I have a quick question. For the following proof why is ##\left ( \frac{n}{n+1}\right )^n < 1##? Proof: We need to prove that, given...

Hello. I'm studying improper integrals in real analysis. However, two problems are very difficult to me. If you are OK, please help me.(heart) 1.2. I have solutions about above problems. However, I don't know how I approach and find the way for solving them.

1) Use mathematical induction to prove that for any k ∈ N, lim (1+k/n)^n = e^k. I already used monotone Convergence Thm to prove k=1 case. Do I just need to go through the same process to show k? If not, could you please help? 2) Suppose that ( x_n ) is a sequence of real numbers, ( y_n...