Real analysis Definition and 509 Threads

Homework Statement Let f : A -> B be a bijection. Show that if a function g is such that f(g(x)) = x for all x ϵ B and g(f(x)) = x for all x ϵ A, then g = f^-1. Use only the definition of a function and the definition of the inverse of a function. Homework Equations The...

Homework Statement Show that the set of all finite subsets of N is a countable set. The Attempt at a Solution At first I thought this was really easy. I had A = {B1, B2, B3, ... }, where Bn is some finite subset of N. Since any B is finite and therefore countable, and since a union of...

Google has my particular homework online. I am doing 1.5.6, 1.5.7, 1.5.8 On 1.5.6 a), I created a function f(x) such that {a} if x = a, {b} if x = b, {c} if x=c. This is 1-1 since each element of A gets mapped to something different. Its obviously not onto. Skipping down to 1.5.7, I need...

Homework Statement Show that if K is compact and F is closed, then K n F is compact. Homework Equations A subset K of R is compact if every sequence in K has a subsequence that converges to a limit that is also in K. The Attempt at a Solution I know that closed sets can be...

Homework Statement What is wrong with my solution?... I don't quite understand where do I go from there... Homework Equations The Attempt at a Solution

Homework Statement The problem and my solution attempt are in the attached file. Am I doing it right? I didn't write the final answer because it is not what I expected. Just wanted to hear if I made any mistakes. Thank you. Homework Equations The Attempt at a Solution

Homework Statement The problem #11. The Attempt at a Solution My partial answer is attached. There, I found E\F. I still don't understand what is f(E) and f(F) and how to derive them from E and F.

Homework Statement The problem and my attempt are attached. I am unable to solve the function for x. Homework Equations The Attempt at a Solution

Homework Statement The problem is attached. Please help me out in understanding this problem. This is not a HW question, just for my own understanding... Homework Equations The Attempt at a Solution

Goal: to show yn=x This particular part of the proof supposes that yn>x. So we want an h>0 such that (y-h)n>x yn-(y-h)n<yn-x yn-(y-h)n=(y-(y-h))(yn-1+yn-2(y-h)+...+(y-h)n-1)<hnyn-1 this yields h=(yn-x)/(nyn-1) my question: how the heck does one derive h from this?

Goal: to show yn=x This particular part of the proof supposes that yn>x. So we want an h>0 such that (y-h)n>x yn-(y-h)n<yn-x yn-(y-h)n=(y-(y-h))(yn-1+yn-2(y-h)+...+(y-h)n-1)<hnyn-1 this yields h=(yn-x)/(nyn-1) my question: how the heck does one derive h from this?

Homework Statement The Attempt at a Solution The solution at the end of the book says that the answer for a) is A5. Why is it so? Please also explain me the meaning for the question b).

Homework Statement What am I asked to do in the problem? Am I just asked to draw a diagram or to prove a) and b)? Homework Equations The Attempt at a Solution

Homework Statement I am trying to understand the definition of Direct and Inverse Images in Real Analysis I from my book, see attachment please.

I am taking Real Analysis I this semester. I am blown away with its difficulty. Proofs are so hard to comprehend, I am at total loss... I am seeking for your advice on how to understand Real Analysis I for a beginner. What kind of learning techniques should I use to comprehend the material...

Homework Statement Let A and B be bounded sets for which there is \alpha > 0 such that |a -b| \geq\alpha for all a in A and b in B. Prove that outer measure of ( A \bigcup B ) = outer measure of (A) + outer measure of (B) Homework Equations We know that outer measure of the union is...

Hi peeps! I was reading Haaser-Sullivan's Real Analysis and came across a problem for which I have a doubt. A part of it is stated like this : " For all x in the closed interval [a,b] in R, |g'(x)|<=1 '' (g(x) is, of course, a real-valued function of a real variable and that's all we know...

Homework Statement Suppose k>2, x, y in R^k, |x-y| = d > 0, and r > 0. Prove if 2r > d, there are infinitely many z in R^k such that |z-x| = |z-y| = r (In Principles of Mathematical Analysis, it is problem 16(a) on page 23.) Homework Equations |ax| = |a||x| |x-z| < or = |x-y| + |y-z|...

simultaneously. Can it be done? How many hours are spent outside of class in each subject?

Homework Statement 1. Let xn and yn be sequences in R with yn+1 > yn > 0 for all natural numbers n and that yn→∞. (a) Let m be a natural number. Show that for n > m \frac{x_n}{y_n} = \frac{x_m}{y_n} + \frac{1}{y_n} \sum_{k=m+1}^{n} (x_k - x_{k-1}) (b) Deduce from (a) or otherwise that...

There is a Harvey Mudd College first semester real analysis course posted at http://www.youtube.com/user/Learnstream, based on the classic text Principles of Mathematical Analysis (Baby Rudin), by Walter Rudin. Professor Francis Su, who delivers these lectures, does a great job helping to tie...

Homework Statement Suppose \sum n converges and an is greater than 0 for all n. Show that the sum of 1/an diverges. Homework Equations The Attempt at a Solution

Suppose that ak is a decreasing sequence and (ak) approaches 0. Prove that for every k in the natural numbers, ak is greater than or equal to 0. I was thinking I should assume the sequence is bounded below by 0 and do a proof by contradiction. Any suggestions?

Homework Statement Please give examples -functions continuous nowhere, continuous at one point – functions differentiable everywhere but with discontinuous derivative – Examples of uniformly continuous functions, functions not uniformly continuous – Combinations of the above. For...

Homework Statement If x and y are arbitrary real numbers. x>y. prove that there exist at least one rational number r satisfying x<r<y, and hence infinitely. The Attempt at a Solution well, I have done my proof, but comparing to the solution offered by...

Homework Statement Let \mathcal{F} \subset C(\mathbb{R}) be a set of continuous functions such that for each x \in \mathbb{R} there is an M_x > 0 such that |f(x)| \leq M_x for all f \in \mathcal{F}. Homework Equations Prove that there is a nonempty open subset Y \subseteq X and an M...

Homework Statement Let f, g be continuous from R to R (the reals), and suppose that f(r) = g(r) for all rational numbers r. Is it true that f(x) = g(x) for all x \in R?Homework Equations The Attempt at a Solution Basically, this seems trivial, but is probably tricky after all. I know that...

Homework Statement Prove that the rationals as a subset of the reals can all be contained in open intervals the sum of whose width is less than any \epsilon > 0. Homework Equations The Attempt at a Solution

Homework Statement I'm trying to show equivalence of two statements: Let f:S-->T be a function, show that f is 1-1 (injective) is equivalent to f(A n B) = f(A) n f(B) for all A,B subsets of S. The Attempt at a Solution I know equivalence means iff, so I started by assuming f is 1-1...

Hi, I have a few questions because I'm watching a lecture on real analysis & I'm a little bit unsure of a few things. I have them in point form for your convenience in answering. http://www.youtube.com/watch?v=lMHR6d0leKA&NR=1 1. (from 2.30 in the video - no need to watch) A & B are sets &...

I have a question about university course offerings. This semester I'm taking a course called "Introduction to Analysis," which uses Edward Gaughan's Introduction to Analysis, and is basically just a more rigorous/proof-based coverage of the topics we learned in the first two semesters of...

Homework Statement Homework Equations In the image above The Attempt at a Solution Well it's a proof. I am thinking about doing it directly. Somehow showing that sup of bn is 2 and inf of bn is 1 and therefore the sequence must be between 1 and 2 for all n. But I am not sure...

Homework Statement Prove that 2^n + 3^n is a multiple of 5 for all odd n that exist in the set of natural numbers. Homework Equations The Attempt at a Solution Suppose the contrary perhaps and do a proof by contradiction? Perhaps induction? edit: done, thank you. please look at second proof :)

Homework Statement Prove: abs(abs(x)-abs(y))<=abs(x-y) Homework Equations Triangle Inequality: abs(a+b)<=abs(a)+abs(b) The Attempt at a Solution This is what i have so far: Let a=x-y and b=y. Then abs(x-y+y) <= abs(x-y)+abs(y) which becomes abs(x)-abs(y)<=abs(x-y). From...

Prove that abs(x-y) < ε for all ε>0, then x=y. I really do not know how to start this... I have tried to do the contra positive which would be If x does not equal y, then there exist a ε>0 such that abs(x-y) >= ε. Can someone help me and lead me to the right direction.

Hey guys, got stuck on this question while doing homework. I would appreciate any help. Let a,b exist in reals. Show that if a<=b1 for every b1 > b. then a <= b. I really got nowhere. I tried letting b1(n)=b+nE where E is a infinitesimal. Then a <= b+nE for all n. Don't really know how to...

Homework Statement Let W\subset S \subset \mathbb{R}^n. Show that the following are equivalent: (i) W is relatively closed in S, (ii) W = \bar{W}\cap S and (iii) (\partial W)\cap S \subset W. Homework Equations The only thing we have to work with is the definitions of open and closed sets...

I recently bought Real Analysis by Haaser and Sullivan. Is this a good introductory real analysis book? I really bought it for fun. I'm not going to put a formal real or complex analysis course in my math minor. Well, I'm on page two at the ordered pair proof. Why is an ordered pair...

what does the function involving rho mean in this definition? (in picture)

Prove or disprove that if m and n are integers such that mn = 1 then either m= 1 & n = 1 or else m = -1 & n = -1.

Prove or disprove that if m and n are integers such that mn = 1 then either m= 1 & n = 1 or else m = -1 & n = -1.

Homework Statement Let Sn be a sequence in R Prove lim Sn= = 0 if and only if lim abs(Sn) = 0 Homework Equations none The Attempt at a Solution I think this is someone ciruclar logic and that is why I am stuck Assume lim Sn = 0, thus for n > N implies |Sn| < epsilon or...

Homework Statement Let f be any function on the real line and suppose that: |f(x)-f(y)|<=|x-y|^2 for all x,y in R. Prove that f is a constant function. Note: "<=" reads "less than or equal to" Homework Equations The Attempt at a Solution I have tried proof by contradiction, it...

1. Royden Chapter 4, # 16, P.94 Establish the Riemann-Lebesgue Theorem: If f is integrable on ( - \infty, \infty) then, \mathop{\lim}\limits_{n \to \infty}\int_{\infty}^{\infty}f(x) \cos nx dx =0 2. The hint says to use this theorem: Let f be integrable over E then given \epsilon...

Homework Statement If a_n diverges to +inf, b_n converge to 0; prove a_n*b_n diverges to +inf Homework Equations The Attempt at a Solution My attempt follows: I seem to have trouble getting things in the right order, so I am trying to work on my technique, with your help. Also...

Homework Statement Let A be a nonempty set of real numbers which is bounded below. Let -A be the set of all numbers -x, where x is in A. Prove that: inf A = -Sup(-A) Homework Equations What should I use as a starting statement? I understand that its true...

Homework Statement Limit of {sn} as n goes to infinity exists provided for all sigma >0 there exists some integer N such that |sn-L| < sigma where n greater than or equal to N. Prove equivalent to alternate definition: limit exists provided that for every positive inteer m there...

Homework Equations Prove the following for n > 1, n is a natural number. Sum (from i=1 to n) of: 1/sqrt(i) is > then sqrt(n) The Attempt at a Solution To be honest I have spent 2+ hours on this with little results. I have tried induction on n, I tried showing one increases more every step...

iam reading analysis by terence tao 2 i can understand the book but cannot express the steps properly while writing proofs and solutions so please suggest me some book which has elaborate steps which can help me in writing whole steps involved in the solution