Real analysis Definition and 524 Threads

I'm a math major, and recently started taking upper level math classes. So far, the only upper levels I've taken in math are Abstract Algebra and Applied Combinatorics. To be honest, I didn't really work as hard as I should have in Abstract, and feel like my proof writing skills are not all...

I was reading a textbook on real analysis and came across this definition:Given a real sequence we say x is an accumulation point if given any \in greater than 0 we can find infinitely many natural numbers n such that |xn-x| is less than \in. I also found a theorem that stated if a real...

Real Analysis will be the most rigorous, proof-based course I've taken for my math major, and I'm concerned because a lot of people at my school HATE the course. Any tips on preparation? Surviving?

Hello all, In the Fall I am planning on taking Real Analysis, Abstract Algebra and doing an independent study in something(my professor has yet to get back to me on what he is willing to do it in). My question is would it be too much of a workload to try and do another independent study in...

I'm preparing to start a year-long sequence of 400-level real analysis using Rudin's Principles of Mathematical Analysis 3E in my second undergrad year, and my advisor recommends I take the graduate-level sequence the following year through Real Analysis by Stein and Shakarchi. Since both of...

Hi, I am thinking of taking Intro to Numerical Analysis and Real Analysis 1 course next semester, but was wondering if maybe that'll be too much of a load? Is numerical analysis a tough course? These courses will be taken alongside 2 other statistics courses and maybe a history class...

Let f : R to R be a continuous function, and assume anti-derivative of f(x)dx from m to n≤ (n-m)^2 for every closed bounded interval [m,n] in R. Prove that f(x) = 0 for all x in R. I tried using fundamental theorem of calculus but got stuck. Any help/suggestion would be appreciated.

Homework Statement limit of the sequence, [xn]=(-3n2+n+1)/(n2-2n+3) Homework Equations I so far know about the definition of a limit, squeeze principle, and lim[xnyn] = 0 if xn or yn goes to 0 The Attempt at a Solution Tried the definition of the limit but the algebra got really...

Homework Statement Give an example of a continuous function f:R^2→R having partial derivatives at (0,0) with f_1 (0,0)≠0,f_2 (0,0)≠0 But the vector (f_1 (0,0),f_2 (0,0)) does not point in the direction of maximal change, even though there is such a direction. (If this is too difficult...

Homework Statement It's not a HW problem. I was reading baby Rudin, in chapter 6 when it talks about Riemann–Stieltjes integral, it claims that given ε>0, we could choose η>0 such that (α(b)-α(a))η<ε. I wonder why it is true. I proposed this question to myself: Suppose that ε>0 is an...

Homework Statement Suppose r>1. Prove the sequence \sqrt[n]{1 + r^{n}} converges and find its limit. Homework Equations The Attempt at a Solution It's obvious that the sequence converges to r, so I know where I need to end up. My first instinct is to use the squeeze theorem...

Homework Statement It is a 4 parter, but i got 3 and 4 done. a) Find f ([0,3]) for the following function: f(x)=1/3 x^3 − x + 1 b) Consider the following function : f(x) = e^(−ax) (e raised to the power of '-a' times 'x') a, x ∈ [0,∞) Find values of a for which f is a contraction ...

Homework Statement a) Find f ([0,3]) for the following function: f(x)=1/3 x^3 − x + 1 b) Consider the following function : f(x) = e^(−ax) (e raised to the power of '-a' times 'x') a, x ∈ [0,∞) Find values of a for which f is a contraction . c) Prove that for all x,y ≤ 0 | 2^x −2^y | ≤ |x−y|

Homework Statement a) Given the definition of the divergence of a sequence {a_n} : "For any H >0 we can find a number NH such that a_n >H, for all n>N_H" prove that {a_n * b} diverges if {a_n } diverges for any b ≠ 0 . b) Find the supremum and infimum for the se… 1 - 1/n } and, if...

Homework Statement a) Show that the series ∑ from n = 1 to infinity 1/n^p where p converges when p > 1 and diverges for p=1. b) Prove that the following series diverges: ∑ from n = 1 to infinity sqrt(n)/n+1 c) Use an appropriate test to show whether ∑ from n = 1 to infinity [(−1)^n *...

Hello there, Is real analysis really important in the process of learning statistics? The reason is, I suck at real analysis(actually, I failed it) and do really good at stats(I am in my third yr. at uni.). Should I continue studying stats or switch to another major? I am looking...

Hi all, Is differential equation a prerequisite to study real analysis (in context of baby Rudin)? And does it have any use in measure theory or Stochastic Calculus? Thanks in advance.

Hi all, Is differential equation a prerequisite to study real analysis (in context of baby Rudin)? And does it have any use in measure theory or Stochastic Calculus? Thanks in advance.

Homework Statement Use the fact that an= a + (an - a) and bn= b + (bn - b) to establish the equality an*bn - ab = (an-a)(bn-b)+b(an-a)+a(bn-b). Then, use this equality to prove that the sequence {an*bn} converges to ab. Homework Equations Definition of convergence: |an*bn - ab| < ε...

Homework Statement Suppose f: ℝ-{0} → ℝ has a positive limit L at zero. Then there exists m>0 such that if 0<|x|<m, then f(x)>0.Homework Equations The definition of the limit of a function at a point is: (already assuming f to be a function and c being a cluster point) A real number L is said...

(typo: title should be Feeling, not Feelings. Whoops) Hi there, I'm a second-year student at a highly ranked private liberal arts school and I am pursuing a BA in Math. Once I graduate, I want to pursue a PhD in either Math or CS. I obviously don't know exactly what I want to focus on, but...

Homework Statement Prove that every infinite subset contains a countably infinite subset. Homework Equations The Attempt at a Solution Right now, I'm working on a proof by cases. Let S be an infinite subset. Case 1: If S is countably infinite, because the set S is a subset...

Homework Statement Suppose that A and B are both countably infinite sets. Prove that there is a one to one correspondence between A and B. Homework Equations The Attempt at a Solution By definition of countably infinite, there is a one to one correspondence between Z+ and A and...

Hello, I'm not quite sure if this kind of question can be posted on this board. Please excuse me if not. I started studying real analysis with Rudin's Principles of Mathematics which was relatively compact. Then I bought Apostol's book which was much more helpful because it was more...

Hi, I'm an undergraduate 4th year Electronics engineering student. So far I have taken courses from various fields of microelectronics and telecommunications. This year, I've decided to direct my career more to telecommunications (might be a field like wireless communications, digital image...

Homework Statement Let S be a non empty set that is bounded about and β = sup S. Prove that for each ε > 0 there exists a point x in S such that x > β - ε.Homework Equations The Attempt at a Solution I don't really know how to begin this. I know it's true; I'm looking at the problem and I'm...

Hi, I am currently taking real analysis(undergraduate course), and am using the book "Introduction to Real Analysis" by Bartle and Sherbert. I think the book is okay overall, but I was hoping to purchase a secondary text to look at whenever I am confused with a proof or something in the...

Let n ≥ 1 be an integer and ε > 0 a real number. Without making reference or use of nth roots, prove that there exists a positive integer m such that \left (1- \frac{1}{m} \right )^{n}> 1-\varepsilon How would I go about proving this? Would I just solve for m?

Homework Statement Let {xn} be a sequence of real numbers. Let E denote the set of all numbers z that have the property that there exists a subsequence {xnk} convergent to z. Show that E is closed. Homework Equations A closed set must contain all of its accumulation points. Sets with no...

I've seen this thread: https://www.physicsforums.com/showthread.php?t=297842 and that is the exact question I need to to answer. What is the radius of convergence of 1/(1+x^2) expanded about x_0=1? The problem is, I can only use an argument in real analysis. I see the answer is...

Homework Statement I attached a .bmp file Homework Equations The Attempt at a Solution I don't get this one at all. Since f is differentiable, the interval [a,b] must be continuous, and I cannot use rational/irrational tricks like in limit problems. does anyone have suggestions?

Suppose we have: f(x)= 1 if 0\leq x \leq 1 AND 2 if 1\leq x \leq 2 Using the definition, show that f is Riemann integrable on [0, 2] and find its value? I have a general idea of how to complete this question using partitions and the L(f,P) U(f,P) definition, but am not quite receiving the...

Let f be a continuous function defined on (a, b). Supposed f(x)=0 for all rational numbers x in (a, b). Prove that f(x)=0 on (a, b). i don't even know where to start...any tips just to point me in the right direction?

Homework Statement Let (u_{n})_{n} be a real sequence such that lim u_{n} = 0 as x→∞ and let (v_{n})_{n} be a bounded sequence. Show that lim (u_{n})_{n}(v_{n})_{n} = 0 as x→∞ Homework Equations The Attempt at a Solution Since (v_{n})_{n} is bounded then it has a least upper bound and...

I'm currently attending university, and I'm comfortable with remembering my math skills. However, I am planning to take Real Analysis 1 in the Spring '12 and then the 2nd course in Spring '13. During that time, I will be taking Advanced Calc, but what do you think? Do you think this will...

Homework Statement Let f(x)=x^4 - x - 1. Show that f(x)=0 has two real roots. Homework Equations None The Attempt at a Solution x(x^3 - 1 - 1/x) = 0 which gives x=0 and x^3 - 1 - 1/x=0, x^2 - 1/x - 1/x^2=0, but WolframAlpha says x~~0.724492 and x~~-1.22074. I kept dividing by x it but...

Homework Statement If the domain of a continuous function is an interval, show that the image is an interval. Homework Equations Theorem from book: f is a cont. function with compact domain D, then f is bounded and there exists points y and z such that f(y) = sup{ f(x) | x ∈ D} and...

HELP! real analysis question: continuity and compactness Homework Statement Let (X,d) be a metric space, fix p ∈ X and define f : X → R by f (x) = d(p, x). Prove that f is continuous. Use this fact to give another proof of Proposition 1.126. Proposition 1.126. Let (X, d) be a metric space...

Do you think it will be too much of a load to take Linear Algebra and Real Analysis in the same semester? Please note that this Linear Algebra course is not an intro course, it's an upper level mathematics course. Will it be similar to the intro to Linear Algebra course I'm taking now? Also, in...

I'm a sophomore math major, and I' currently taking proofs, linear algebra (not proofs-based), and calc 3. These classes aren't that bad so far. I met with a math adviser today, and he told me for my major requirements I should take real analysis 1&2, Linear algebra, and abstract algebra for a...

The problem statement Let f:[a,b]→\mathbb{R} be differentiable and assume that f(a)=0 and \left|f'(x)\right|\leq A\left|f(x)\right|, x\in [a,b]. Show that f(x)=0,x\in [a,b]. The attempt at a solution It was hinted at that the solution was partly as follows. Let a \leq x_0 \leq b. For all x\in...

So, I am taking introduction to real analysis next semester, and I heard that it may be a challenging course. But what I want to know is the applications of real analysis, which I imagine there are some of, considering how the phrase "real analysis" seem to be thrown around quite a bit. Thanks.

Homework Statement Suppose {Xn}, {Yn} are sequences in ℝ and that |Xn-Yn|→0. Show that either: a) {Xn} and {Yn} are both divergent or b) {Xn} and {Yn} have the same limit. Homework Equations N/A The Attempt at a Solution I first prove that lim(Xn-Yn)=lim(Xn)-lim(Yn). I am not...

Homework Statement > a[1], a[2], a[3], .. , a[n] are arbitrary real numbers, prove that; abs(sum(a[i], i = 1 .. n)) <= sum(abs(a[i]), i = 1 .. n) Homework Equations The Attempt at a Solution I have uploaded my attempt as a pdf file, since I'm not too familiar with the...

Homework Statement Prove that br+s=brbs if r and s are rational. Homework Equations So far we know the basic field axioms and a a few other things related to powers. 1.) For every real x>0 and every integer n>0 there is one and only one positive real y such that yn=x 2.) if a and b...

Homework Statement Let f:Rnxn-->Rnxn be defined by f(A) = A2. Prove that f is differentiable. Find the derivative of f. Homework Equations f(a + h) = f(a) + f'(a)h + \phi(h) The Attempt at a Solution f(A + H) = (A + H)2 = A2 + AH + HA + H2 f(A) is given by A2. So the sum of...

Anyone interested in opening online study group on Real Analysis? I want to use https://www.amazon.com/gp/product/0486469131/?tag=pfamazon01-20 for the study group. Method: Some time will be given for self study then, group will discuss concepts and solve exercises from the book. [each...

Let f and g be functions such that (g\circf)(x)=x for all x \epsilonD(f) and (f\circg)(y)=y for all y \epsilonD(g). Prove that a g = f^-1 Pf/ How would you go about starting this besides saying Let f and g be functions such that (g\circf)(x)=x for all x \epsilonD(f) and (f\circg)(y)=y for...

Homework Statement Find the supremum of E=(0,1) Homework Equations The Attempt at a Solution By definition of open interval, x<1 for all x in E. So 1 is an upper bound. Let M be any upper bound. We must show 1<=M. Can I just say that any upper bound of M must be greater than or...

Homework Statement If a < b-\epsilon for all \epsilon >0, then a<0 Homework Equations All I really have are the field axioms of the real numbers and the order axioms (trichotomoy, transitive, additive property, multiplication property). The Attempt at a Solution Well I broke this...