Proof Definition and 999 Threads

If i take any two APs say a=1; d=5; 1,6,11,16,21,26,31,36,41,46,51......1+(n-1)5. say a=2; d=3; 2,5,8,11,14,17,20,23,26,29,32,35,38,41......2+(n-1)3. If i pick out the common terms here, I get an AP again o common difference 15. 11,26,41....11+(n-1)15 How can i prove that the common terms of...

What is the probability that a randomly chosen natural number, which is not smaller than a given natural number x, is greater than x? I think it must be ##1## because there are infinitely numbers that greater than ##x## but how can I proof this? Thank you.

Hello, My name is Nikolaos Bafitis, and currently I am student at St. Johns College Highschool in DC. Two years ago I moved back from Italy where I stayed in for 3 years. Upon my return to the US I have noticed that I seem to be getting slower at math skills and speed after being placed in...

To prove the upper bound: Let ##c>0##, divide it into ##f(x)## and the coefficients in the final line of the synthetic division tableau are all non-negative. Thus ##f(x)=(x-c)q(x)+r##, where ##r \geq 0## (since the coefficients are given as all non-negative) and is a constant because it's degree...

Hi, PF Theorem The set ##\mathbb{Q}## of the rational numbers is countably infinite. Proof The rational numbers are arranged thus...

Hello! I saw such interpretation of principle of relativity when I read proof of invariance of infinitesimally small interval: "The second inertial frame of reference looks from the first in no way different from how the first inertial frame of reference looks from the second." Proof of...

So far, I've got that ## g(a)=(c-1)a+\frac{3}{4}a^3\implies g'(a)=c-1+\frac{9}{4}a^2 ##. I know that if the first derivative of a function is positive (greater than ## 0 ##), then that function is always/strictly increasing. However, how should I construct this proof in order to show that...

So using the multiplicative inverse axiom we have : 1) x . x^-1 = 1 2) x . (1/x) = 1 I have no idea why do mathematicians define the multiplicative inverse of a number x to be the "fraction" 1/x. But I know for sure that multiplying any number a for example by the multiplicative inverse of x is...

My solution is: Let ##\lim_{ x \to 0}\left(\dfrac {1}{\sin^2x}-\dfrac1{x^2}\right)=L## Let ##x=2y## ##\lim_{ y \to 0}\left(\dfrac {1}{\sin^22y}-\dfrac1{4y^2}\right)=\lim_{ y \to 0}\left(\dfrac1{4\sin^2y\cdot\cos^2y}-\dfrac1{4y^2}\right)=L## ##=\dfrac14\lim_{ y \to...

My solution: Assume it is reducible, i.e., ##a_n X^n + ... + a_0 = (b_k X^k + ...+ b_0)(c_m X^m+ ... +c_0)##. ##a_0=b_0 c_0##. Since ##p \mid a_0##, either ##p \mid b_0## or ##p \mid c_0##, but not both, because ##p^2 \nmid a_0##. Assume ##p \mid b_0, p \nmid c_0##. ##a_1=b_0 c_1+b_1 c_0##. ##p...

Consider the following proof: My question is, does it in fact use induction? It says, "Assume now that the theorem is true for k-1 elements...," but I don't think it uses this assumption to prove that it is true for k elements, which would be an induction step.

Iv'e got this proof from some claim: Any Pure maths doctors out there who can explain to me why since ##V\ne \emptyset## that ##1\notin \mathfrak{b}##? Thanks!

Proof: Consider the nonlinear integro-differential equation ## \frac{dx}{dt}=-\lambda x(t)+\epsilon x(t)\int_{0}^{\infty}f(t-s)x(s)ds, \lvert \epsilon \rvert<<1, x(0)=A ##, where ## \lambda ## is a positive constant and ## f(z) ## is a sufficiently well-behaved function. Let ## \epsilon=0 ##...

See the attached image for my attempt. My main concern is can I assume that y > x prove it for that case and then show it is equal if y = x. My whole proof is centered around y > x so if i cannot make that assumption then I have to start over. Let me know your thoughts. Thanks in advance for...

Hello! I read Landau & Lifshitz' Classical Theory of Fields [Link to copyrighted textbook redacted by the Mentors] (see pic below) and I was confused when I saw in proof that coefficient "a" between spacetime interval (ds)2 and (ds')2 can only depend on the absolute relative velocity between...

Here's an excerpt from the proof of the change of variables formula in Folland's book (Theorem 2.47, page 76, 2nd edition, 6th and later printings): For reference, see Theorem 2.40 below. I don't understand how he is using Theorem 2.40 in the quoted passage. Which part of Theorem 2.40 is he...

I am self-learning analysis. My steps are as follows, For any ##ε>0##, there is a ##\delta>0## such that, ##|(x^3+2x^2-4x-8) -0|<ε## when ##0 < |x-2|<\delta## Let ##\delta≤1## then ##1<x<3, x≠2##. ##|(x^3-2x^2-4x-8) -0|=|(x-2)(x+2)^2|=|x-2||(x+2)^2| <\delta |(x+2)^2|<19\delta## Taking...

From Blundell and Blundell Chapter 20 Problem 20.3. I have proved that $$1-e^{-\beta \omega}=2\sinh\left(\frac{\beta \omega}{2}\right)$$ with no problem, but I am stuck on the ##\coth## term. I have tried to solve this but it gets messy and I'd rather not include them here. Thanks!

I attached my attemp at the solution. I am trying to start with continuity at 0 and end up with limit of f(x) equals f(c) as x goes to c. Could someone take a look at the attached image and let me know if I am on the right track or where I went astray Sorry picture is rotated I tried but...

I was intrigued by a comment in Brilliant.org: Besides the proof provided by Brilliant, I also found a couple of other websites. But none of these proofs were entirely clear to me. So I tried to come up with my own proof. Since I am not a group theorist, I wanted to ask if the proof makes...

For g) how should I argue this claim? To me it seems straight forward because u is linearly related to z. Then so do their derivatives. And by the description of the model, ##\dot z## is linear to y. So it's quite obvious but not sure what I should pay more attention to when I write my proof...

Proof: Consider the transformation ## x=\frac{1}{\sqrt{1+e^{-2q}}} ## and ## y=\frac{1}{\sqrt{1+e^{-2p}}} ## with the Hamiltonian function ## H(q, p)=ap-b\cdot ln(e^{p}+\sqrt{1+e^{2p}})+cq-d\cdot ln(e^{q}+\sqrt{1+e^{2q}}) ##. Let ## \dot{x}=\frac{dx}{dt}=(a-by)x(1-x^2)=(a-by)(x-x^3) ## and ##...

I just decided to look at Landau & Lifshitz' Classical Theory of Fields (English version, 4th ed), and I am a bit embarrassed to be confused already on page 4&5 of this book. The book can be viewed on archive.org. The goal of this section of the book is to show ##s = s'## starting from only the...

I was looking at the proof of zeroth law of thermodynamics from the original paper by Bardeen, Carter, Hawking, which can be found here. Now, we have the Killing vector which is the generator of the horizon, we call it ##l^\mu##, and auxiliary null vector field ##n^\mu##, which we define to be...

https://www.quantamagazine.org/monumental-proof-settles-geometric-langlands-conjecture-20240719 https://people.mpim-bonn.mpg.de/gaitsgde/GLC/ https://en.wikipedia.org/wiki/Langlands_program https://www.quantamagazine.org/what-is-the-langlands-program-20220601/...

Hello! I have some troubles diving in the proof of this lemma Lemma. Let ##S## be locally compact, Hausdorff and second countable. Then every open cover ##\lbrace U_\alpha \rbrace## of ##S## has a countable, locally finite refinement consisting of open sets with compact closures. Proof...

Proving this geometrically [1] gives ##J = r.## Why is the ##-r## one wrong? Why is ##(x, y) \rightarrow (\theta, r)## is different from ##(x, y) \rightarrow (r, \theta)##? Edit: In Paul's Notes [2] it seems like ##J## is always positive, but online says it can be negative... [1] The first...

I'm reading "Complex Made Simple" by David C. Ullrich and here i have a problem with the proof of a theorem: Theorem Suppose that ##p : X \to Y## is a covering map. If ##\gamma : [0,1] \to Y## is continuous, ##x_0 \in X## and ##p(x_0) = \gamma(0)## then there exists a unique continuous function...

In the photos are two proof questions requiring proving convergence of sequence from convergent subsequences. Are my proofs for these two questions correct? Note in the first question I have already proved that f_n_k is both monotone and bounded Thanks a lot in advance!

For this problem, My proof is Since ##f'## is increasing then ##x < y <z## which then ##f(x) < f(y) < f(z)## This is because, ##f''(t) \ge 0## for all t ## \rightarrow \int \frac{df'}{dt} dt \ge \int 0~dt = 0## for all t ##\rightarrow \int df' \geq 0## for all t ##f ' \geq 0## for all t...

For this problem, I'm confused by the implication from the antecedent ##0 < |x - c| < \delta## to the consequent. Should the consequent not be ##|f''(x) - f''(c)| < \frac{1}{2}## where ##\epsilon = \frac{1}{2}## (Since we are applying the definition of a limit for the first derivative curve)...

For this problem, My solution is If ##c < 1##, then let a be a number such that ##c < a < 1 \implies c < a##. Thus for some natural number such that ##n \geq N## ##|x_n|^{\frac{1}{n}} < a## is the same as ## |x_n| < a^n## By Law of Algebra, one can take the summation of both sides to get...

For this problem, My solution is, ##F(x)=\left\{\begin{array}{ll} e^{-\frac{1}{x}} & \text { if } x>0 \\ 0 & \text { if } x \leq 0\end{array}\right.## The we differentiate both sub-function of the piecewise function. Note I assume differentiable since we are proving a result that the function...

For (a) and (b), Does someone please know how to prove this? I don't have any ideas where to start. Thanks!

If ##V## is timelike Killing with Frobenius condition ##V_{[\alpha} \nabla_{\mu} V_{\nu]} = 0## then you can derive the equation:$$\nabla_{\mu} (|V|^2 V_{\nu}) - \nabla_{\nu} (|V|^2 V_{\mu}) = 0$$which has the solution$$V_{\alpha} = \partial_{\alpha} \phi \quad \mathrm{where} \quad \phi = x^0 +...

Hi, I don't know if I have solved task correctly I used the epsilon-delta definition for the proof, so it must hold for ##f,g \in (C^0(I), \| \cdot \|_I)## ##\sup_{x \in [a,b]} |F(x)-G(x)|< \delta \longrightarrow \quad |\int_{a}^{b} f(x)dx - \int_{a}^{b} g(x)dx |< \epsilon## I then...

I have, Using ##\ cosh 2x = 2 \cosh^2 x - 1## ##\cosh x = 2 \cosh^2\dfrac{x}{2} -1## Therefore, ##\cosh x -1 = 2 \cosh^2\dfrac{x}{2} -1 - 1## ##\cosh x -1 = 2 \cosh^2\dfrac{x}{2} -2## ##=2\left[ \cosh^2 \dfrac{x}{2}...

For this true or false problem, My solution is, With rearrangement ##\frac{f(x) - f(a)}{x - a} > f'(a)## for ##x < a## since ##f''(x) > 0## implies ##f'(x)) > 0## from integration. ##f'(x) > 0## is equivalent to ##f(x)## is strictly increase which means that ##\frac{f(x) - f(a)}{x - a} > f'(a)...

For this problem, I am trying to prove that this function is non-differentiable at 0. In order for a function to be non-differentiable at zero, then the derivative must not exist at zero ##⇔ \lim_{x \to 0} \frac{f(x) - f(0)}{x - 0}## does not exist or ##⇔ \lim_{x \to 0^-} \frac{f(x) - f(0)}{x...

For this problem, THe solution is, However, does someone please know why from this step ##-1 \leq \cos(\frac{1}{x}) \leq 1## they don't just do ##-x \leq x\cos(\frac{1}{x}) \leq x## from multiplying both sides by the monomial linear function ##x## ##\lim_{x \to 0} - x = \lim_{x \to 0} x= 0##...

We consider base case (##n = 1##), ##B\vec x = \alpha \vec x##, this is true, so base case holds. Now consider case ##n = 2##, then ##B^2\vec x = B(B\vec x) = B(\alpha \vec x) = \alpha(B\vec x) = \alpha(\alpha \vec x) = \alpha^2 \vec x## Now consider ##n = m## case, ##B^m\vec x = B(B^{m - 1}...

I have a doubt about this problem. (a) Show that a matrix ##\left(\begin{array}{ll}e & g \\ 0 & f\end{array}\right)## has determinant equal to the product of the elements on the leading diagonal. Can you generalize this idea to any ##n \times n## matrix? The first part is simple, it is just ef...

I am trying to solve (a) and (b) of this tutorial question. (a) Attempt: Since ##c'## is at ##c'(0) = 1##, then from the definition of continuity at a point: Let ##\epsilon > 0##, then there exists ##d > 0## such that ##|x - 0| < d \implies |c'(x) - c'(0)| < \epsilon## which is equivalent to...

Consider this proof: Is it a valid proof? When we divide by ##z##, we assume that ##z \neq 0##. So, we cannot put ##z=0## on the next step. IOW, after dividing by ##z## we only know that $$c_1+c_2z+c_3z^2+...=d_1+d_2z+d_3z^2+...$$ in a neighborhood of ##0## excluding ##0##.

For this problem, The solution is, However, does someone please know why this did not use ##2n ≤ 2n^2 + 2n + 1## which would give ##\frac{3n - 1}{2n^2 + 2n + 1} ≤ \frac{3n}{2n} = \frac{3}{2}##? In general, after solving many problems, it seems that when proving the convergence of a rational...

The problem and solution are, However, I am confused how the separation vector between the two masses is ##\vec x = x \hat{k} = x_2 \hat{x_2} - x_1 \hat{x_1}= l\theta_2 \hat{x_2} - l\theta_1 \hat{x_1 } = l(\theta_2 - \theta_1) \hat{k}##. where I define the unit vector from mass 2 to mass 1...

I am trying to understand the proof given in Ethan Bloch's book "The real numbers and real analysis". I am posting snapshot of the proof in the book. I am also posting theorem 1.2.9 given in the book. Here author is trying proof by contradiction. First, I don't understand why specific...

Hello everyone, I've been trying to understand the proof for the binomial theorem and have been using this inductive proof for understanding. So far the proof seems consistent everywhere it's explicit with the pattern it states, but I've started wondering if I actually fully grock it because I...

Hello, found this proof online, I was wondering why they defined r_2=r_1-(r_1^2-2)/(r_1+2)? i understand the numerator, because if i did r_1^2-4 then there might be a chance that this becomes negative. But for the denominator, instead of plus 2, can i do plus 10 as well? or some other number...