Real numbers Definition and 214 Threads

Homework Statement Let f:R\rightarrowR satisfy f(x+y) = f(x) + f(y) for real numbers x and y. If we let f be continuous, show that \exists a real number b such that f(x) = bx. Homework Equations n/a The Attempt at a Solution Nooooo clue!

Homework Statement show whether the following set of vectors M = \left\{\left(a_{1},a_{2},a_{3}\right) with a_{1},a_{2},a_{3} \in \Re\right\} with the following limitations: 1) a1 is rational 2) a1 = 0 3) a1 + a2 = 0 4) a1 + a2 = 1 is a vector space over the field of real numbers. Homework...

I got this out of An Imaginary Tale: The Story of Sqrt(-1). In section 1.5 of the book, the author explains that Bombelli took x3 = 15x + 4 and found the real solutions: 4, -2±sqrt(3). But if you plug the equation into the Cardan forumla you get imaginaries...

Homework Statement Show that: Let S be a subset of the real numbers such that S is bounded above and below and if some x and y are in S with x not equal to y, then all numbers between x and y are in S. then there exist unique numbers a and b in R with a<b such that S is one of the...

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Homework Statement so.. let the operation * be defined as x*y = x + y + xy for every x,y ∈ S, where S = {x ∈ R : x ≠ -1}. Now i have proven associativity, existence of an identity and inverses, all without commutativity, but i must show that this is an abelian group, so now i have to show...

Homework Statement If f is continuous and f(x)=0 for all x in a dense subset of the real numbers, then f(x)=0 for all x \in \mathbb{R}. Homework Equations N/A The Attempt at a Solution Does this solution work? And if it does, can it be improved in some way? Proof: From the...

Homework Statement Prove that any well-ordered subset (under the natural order) of the real numbers is countable. Homework Equations None. The Attempt at a Solution My attempt thus far has been to prove by contradiction. I didn't see a very clear way to get from well-ordered subset...

I am struggling to understand the proof for integer parts of real numbers. I have used to mean less than or equal to because I could not work out how to type it in. I need to show that: ∃ unique n ∈ Z s.t. nx<n+1 The proof given is the following: Let A={k∈Z : kx} This is a...

How would you solve this without using decisional blocks?

For all positive real numbers x,y prove that: \frac{1}{1+\sqrt{x}}+\frac{1}{1+\sqrt{y}} \geq \frac{2\sqrt{2}}{1+\sqrt{2}}

1. Homework Statement A and B are real numbers satisfying 2(sinA+cosB)sinB=3-cosB. Find 3tan^2A+4tan^2B 2. Homework Equations Trig Identities 3. The Attempt at a Solution well, i expanded 3tan^2A+4tan^2B= 3sin^2A*cos^2B+4sin^2B*cos^2A all over cos^2A*cos^2B after that I am stuck :x

Homework Statement if the real numbers x,y,z,w satisfy (x2/(n2-1))+(y2/(n2-32))+(z2/(n2-52))+(w2/(n2-72)) for n=2,4,6,8 then prove x2+y2+z2+w2=36 Homework Equations The Attempt at a Solution unable to think of anything?:confused:

Homework Statement Suppose that a, b, c are real numbers and x, y, z >= 0. Prove that \frac{a^2}{x} + \frac{b^2}{y} + \frac{c^2}{z} \geq \frac{ (a+b+c)^2}{x+y+z}Homework Equations Cauchy-Schwarz and Arithmetic Geometric Mean inequalities.The Attempt at a Solution I wasn't really sure how to...

If we were to take any two integers on a real number line and mark a point (a number) halfway between the two, do the same in the range between the halfway point and each of the two numbers, and continue the process, would we be able to define all real numbers between the two integers (including...

Homework Statement This is the question followed my my attempt at the solution: Just wondering if this looks right? Thanks for any and all feedback, Jim

Homework Statement Let x and y be real numbers with x<y and write an inequality involving a rational number p/q capturing what we need to prove. Multiply everything in your inequality by q, then explain why this means you want q to be large enough so that q(y-x)>1 . Explain how you...

Homework Statement Homework Equations N/A The Attempt at a Solution Assuming the truth of part a, I proved part b. But now I have no idea how to prove parts a & c. Part a seems true intuitively. The sqaure root of a number between 0 and 1 is will be larger than that number, and if...

Homework Statement I understand everything except the last two lines. I am really confused about the last two lines of the proof. (actually I was never able to fully understand it since my first year calculus) I agree that if ALL three of the conditions n≥N, k≥K, and nk≥N are satisfied...

For 0<x<y, show that x<\sqrt{xy}<1/2(x+y)<y I have no difficulty showing that x<\sqrt{xy} and 1/2(x+y)<y. I am having difficulty with \sqrt{xy}<1/2(x+y). x<y xx<xy x^{2}<xy x<\sqrt{xy} and x<y x+y<y+y x+y<2y 1/2(x+y)<y

Does anyone have or know of any good books that cover the construction of the real numbers via cauchy sequences? I would appreciate any recommendations. Thanks!

Hi, Are these correct? Homework Statement a.) Given that x > y, and k < 0 for the real numbers x, yand , show that kx < ky. b.) Show that if x, y ∈ R, and x < y , then for any real number k < 0,kx > ky 2. The attempt at a solution a.) kx > y...1 x > y x - y is +ve...2 k...

Homework Statement x and y are real numbers. prove that if x2-4x=y2-4y and x not equal to y, then x+y=4. Homework Equations n/a The Attempt at a Solution I tried using cases and making x and y positive and negative or even and odd, and that didnt work. then i tried completing the...

Homework Statement Prove that (-1,1) is homeomorphic to R (real numbers), with the topology given by the usual metric. Homework Equations None. The Attempt at a Solution I constructed the function f(x) = [1/(1-x) - 1/(1+x)]/2 = x/[(1+x)(1-x)] which is continuous and maps (-1,1)...

if a set A is both open and closed then it is R(set of real numbers) how we may show it in a proper way

Hallo, My teacher wrote that: "The set has no interior points, and neither does its complement, R\Q" where R refers real numbers and Q is the rationals numbers. why can't i find an iterior point? thanks, Omri

Hello,I need some advice on a problem. Let f,g:R\rightarrow R (where R denotes the real numbers) be two continuous functions, assume that f(x) < g(x) \forall x \neq 0 , and f(0) = g(0).Define A = \left\{(x,y)\neq (0,0): y< f(x),x \in R\right\} B = \left\{(x,y)\neq (0,0): y> g(x),x \in...

Homework Statement A subset U \subseteq R is called open if, for every x \in U, there is an open interval (a, b) where x \in (a, b) \subseteq U. (a) Show that, in the above dedefinition, the numbers a, b may be taken as rational; that is, if x \in U, there is an open interval (c, d)...

From what I understand, the class of all real numbers that we can represent as a sentence in logic is countable. But I'm not sure if it's a set under the standard ZF axioms... it seems intuitive that it should be, since the axioms are really designed to prevent problems involving sets that are...

Homework Statement For real numbers x(1), x(2), ..., x(n), prove that |x(1) + x(2) +...+x(n)| <= |x(1)|+...|(n)|Homework Equations The Attempt at a SolutionMaybe begin with prooving that x <= |x| ? I am not sure how to do this though. Any help or hints would be great, as I am really stuck on this.

Homework Statement Prove the intervals of real numbers (1,3) and (5,15) have the same cardinality by finding an appropriate bijective function of f:(1,3) ->(5,15) and verifying it is 1-1 and onto Homework Equations I know there are multiple ways to prove one to one and onto I am not sure...

question: let x_1,...,x_n positive real numbers. prove that \lim_{p\to \infty}\left(\frac{x_1^p+...+x_n^p}{n}\right)^{1/p}=max\{x_1,...,x_n\} can you give me some hints ?

I have been trying the proof of the reverse triangle inequality for n+1 real numbers: |x-y1-y2-y3-...-yn| \geq | |x| - |y1| - |y2| - |y3| - ...-|yn| | I know the proof of the reverse triangle inequality for 2 real numbers and the triangle inequality for n numbers. can somebody help ?

Find a set A (subset of R,set of real numbers) and an element a of R such that there is no bijecton from a+A(we add a to the set A)to A. I can't find a good example. Can someone help Are we done if we choose the empty set? (And is the empty set a subset of R?) Thank you

Does there exist a set X\subset\mathbb{R} that has a property m^*(X\cap [0,x]) = \frac{x}{2},\quad\quad\forall x>0, where m^* is the Lebesgue outer measure? My own guess is that this kind of X does not exist, but I don't know why. Anybody knowing proof for the impossibility of this X?

My analysis text mentions in passing that the real numbers can be constructed rigorously starting from set theory. I was wondering if there were a resource on the web that might go over this and show the proofs of how this is done?

Please tell me one of the bases for the infinite dimenional vector space - R (the set of all real numbers) over Q (the set of all rational numbers). The vector addition, field addition and multiplication carry the usual meaning.

[SOLVED] Proof Theory for all real numbers Homework Statement If a and b are real numbers, we define max {a, b} to be the maximum of a and b or the common value if they are equal. Prove that for all real numbers d, d1, d2, x, If d = max {d1, d2} and x ≥ d, then x ≥ d1 and x ≥ d2...

1. Let A be a nonempty set of real numbers which is bounded below. Let -A be the set of numbers -x , where x \in A . Prove that \inf(A) = -\sup(-A) . Intuitively this makes sense if you draw it on a number line. But I am not sure how to formally prove it.

hello, first: excuse me if the question is stupid but I am still at school. my question: some days ago i came across imaginary numbers. You know what I mean - the imaginary number i^2=-1 and the imaginary numberline is not on the the same line as the other numbers. The imaginary numberline is...

hey i can't figure this out: if p and q are positve real numbers and 1/p +1/q = 1 show that if u and v are greater than or equal to zero then uv=< (u^p)/p +(v^q)/q.

There is a theorem (the "Borel lemma") that says: Let (A_n) by any sequence of real numbers. We can built a function "F", indefinitely differentiable, such that if G is the n-derivative of f, G(0) = a_n. Does someone knows a proof or where can I find it? The theorem appears in wikipedia...

Can someone please tell me why the domain of y=2x^2+12x+12 is x elementof IR

Homework Statement Given x<y for some real numbers x and y. Prove that there is at least one real z satisfying x<z<y Homework Equations This is an exercise from Apostol's Calculus Vol. 1. The usual laws of algebra, inequalities, a brief discussion on supremum, infimum and the...

I'm doing self study out of Apostol's Calculus vol. I and I got stuck trying to prove what the author writes is easy to verify, but I can't get my head around it. Basically, this is the problem statement from page 31, last paragraph: Given a positive real number x, let a0 denote the largest...

the problem statement is: if a,b,c are real numbers such that \frac{1}{a+1} + \frac{1}{b+1} + \frac{1}{c+1} = 2 we have to prove that: \frac{1}{4a+1} + \frac{1}{4b+1} + \frac{1}{4c+1} \geq 1 thanks in advance.

Homework Statement If the domain of f is restricted to the open interval (-pi/2,pi/2), then the range of f(x) = e^(tanx) is A) the set of all reals b the set of positive reals c the set of nonnegative reals d R: (0,1] e none of these (from barron's How to prep for ap calc) Homework...

A trivial, yet difficult question. How would one prove that the real numbers are not compact, only using the definition of being compact? In other words, what happens if we reduce an open cover of R to a finite cover of R? I let V be a collection of open subset that cover R Then I make the...

How does one prove this statement? I have no idea how to start. Can someone help? Maybe it has something to do with that Cauchy-Schwartz inequality?

Hey everyone! This question came up in my analysis assignment, we're studying continuity and differentiability at the moment so I'm unsure of my answer! It seems too short :( For a), it seems to follow immediately, If mq = np then m = \frac{np}{q} and so we have (x^{\frac{1}{n}})^m =...