Proof Definition and 999 Threads

A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases. An unproven proposition that is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work.Proofs employ logic expressed in mathematical symbols, along with natural language which usually admits some ambiguity. In most mathematical literature, proofs are written in terms of rigorous informal logic. Purely formal proofs, written fully in symbolic language without the involvement of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics, oral traditions in the mainstream mathematical community or in other cultures. The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.

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  1. L

    Stumped on mathematical proof

    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!
  2. T

    Prove Continuity From Precise Definition of Limit

    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...
  3. E

    I Fundamental theorem of arithmetic from Jordan-Hölder theorem

    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...
  4. S

    Nonlinear dynamics problem from Strogatz textbook

    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...
  5. M

    How should I show that the transformation makes the system Hamiltonian?

    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 ##...
  6. T

    I Help understanding proof of ds=ds' in Classical Theory of Fields

    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...
  7. Antarres

    A Zeroth law of black hole thermodynamics

    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...
  8. D

    B Spherical Lens Model and Proof

  9. Astronuc

    I Monumental Proof Settles Geometric Langlands Conjecture

    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/...
  10. K

    I Tough lemma on locally finite refinement

    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...
  11. L

    Why is the Jacobian for polar coordinates sometimes negative?

    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...
  12. M

    A Question about existence of path-lifting property

    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...
  13. L

    Proving convergence of sequence from convergent subsequences

    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!
  14. M

    Proof given ##x < y < z## and a twice differentiable function

    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...
  15. M

    Differentiable function proof given ##f''(c) = 1##

    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)...
  16. M

    Root test proof using Law of Algebra

    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...
  17. M

    Proving piecewise function is k-differentiable

    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...
  18. M

    Ratio test proof

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

    I Final part of proof of timelike Killing w/ Frobenius --> static

    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 +...
  20. L

    Continuous functions on metric spaces

    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...
  21. chwala

    Prove the given hyperbolic trigonometry equation

    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}...
  22. M

    True or false problem for double differentiable function

    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)...
  23. M

    Non-Differentiable Function proof

    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...
  24. M

    Proving function discontinuous at zero

    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##...
  25. T

    Eigenvalue of matrix proof by induction

    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}...
  26. T

    Linear Algebra Determinant proof

    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...
  27. T

    Medium Hard continuity proof Tutorial Q6

    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...
  28. Hill

    B Identity Theorem for power series

    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##.
  29. M

    Proving convergence of rational sequence

    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...
  30. M

    Coupled pendulum-spring system

    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...
  31. I

    Proving well ordering principle from Peano Axioms

    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...
  32. C

    B I've been trying to understand the proof for the binomial theorem

    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...
  33. L

    I Supremum of a set, relations and order

    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...
  34. M

    Proof that T is bounded below with ##inf T = 2M##

    My first solution is Let ##S = \{x_1, x_2, x_3, ..., x_n\}## ##T = \{2x_1, 2x_2, 2x_3, ... 2x_n\}## ##T = 2S## Therefore, ##inf T = inf 2S = 2inf S = 2M## May someone please know whether this counts as a proof? My second solution is, ##x ≥ M## ##2x ≥ 2M## ##y ≥ 2M## (Letting y = 2M) Let...
  35. M

    Can You Solve a Problem Using the Definition of Supremum?

    For this problem, My solution: Using definition of Supremum, (a) ##M ≥ s## for all s (b) ## K ≥ s## for all s implying ##K ≥ M## ##M ≥ s## ##M + \epsilon ≥ s + \epsilon## ##K ≥ s + \epsilon## (Defintion of upper bound) ##K ≥ M ≥ s + \epsilon## (b) in definition of Supremum ##M ≥ s +...
  36. C

    B Question about the fundamental theorem of calculus

    Hello everyone, I've been brushing up on some calculus and had some new questions come to mind. I notice that most proofs of the fundamental theorem of calculus (the one stating the derivative of the accumulation function of f is equal to f itself) only use a limit where the derivative is...
  37. M

    I Question about branch of logarithms

    I've read a proof from Complex Made Simple (David C. Ullrich) Proposition 4.3. Suppose that ##V## is an open subset of the plane. There exists a branch of the logarithm in ##V## if and only if there exists ##f \in H(V)## with ##f'(z) = \frac{1}{z}## for all ##z \in V##. Proof: One direction is...
  38. A

    Number of Multiplications in the FFT Algorithm

    Hello everyone, maybe some of you know the formula for the number of multiplications in the FFT algorithm. This is again given as ##N/2 \cdot log(N)##. Why is that so? Can you really "prove" this? I can only deduce this from what I know, because we have ##log(N)## levels and ##N/2##...
  39. C

    B Question about change of variables

    Hello everyone, I found a good proof for the area of a circle being ##{\pi}r^2## but I was recently working on my own proof and I used a change of variables and was wondering if I did it correctly and why a change of variables seems to work. I start with the equation of a circle ##r^2 = x^2 +...
  40. fluidistic

    Zero K proof that a chess position contains a checkmate

    Hi people, It's been years I wanted to post this question here. I would like to build a zero knowledge proof that a given chess position contains at least one checkmate. I know that anything provable admits a zero k proof. I know about...
  41. MAXIM LI

    I Determination of error in interpolating polynomial

    Professor showed this result in the lecture without giving any proof (after proving the existence of the interpolating polynomial in two variables). I've been trying to prove it myself or find a book where is proved but I failed. This is the theorem: Let $$ x_0 < x_1 < \cdots < x_n \in [a, b]...
  42. jedishrfu

    B What theorems are available when using modulo arithmetic?

    I'm looking for theorems related to using modulo arithmetic. As an example, if I apply a sequence of arithmetic operations to a given number to get an answer and then apply a modulo operation on the result to get a remainder in a given base. Wiil that be the same if I apply the modulo operation...
  43. G

    The Divergence of the Klein-Gordon Energy-Momentum Tensor

    I've tried this problem so, so, so so so many times. Given the equations above, the proof starts easily enough: $$\partial_\mu T^{\mu\nu}=\partial_\mu (∂^μ ϕ∂^ν ϕ)-\eta^{\mu\nu}\partial_\mu[\frac{1}{2}∂^2ϕ−\frac{1}{2}m^2ϕ^2]$$ apply product rule to all terms $$=\partial^\nu \phi \cdot...
  44. Delta2

    How Does Integration by Factors Relate to the Product Rule and FTC?

    I tried to prove this but I fall into a loop when I try to apply integration by factors, that is I prove that the integral is equal to itself. Any helpfull tips?
  45. BugKingpin

    I Bloch Analysis proof of Theorem 2.5.5 (Definition by recursion)

    Want to understand how set C contains ##N## x H. H is only defined to be a set with element e and as the domain/range of function k. Is this enough information to conclude that the second set in the cartesian product W is H and not a subset of H? My thinking is to show that ##N## and H satisfy...
  46. graphking

    I A strict proof of "why the Earth is a ball"

    "bubbles are ball" is called isoperimetric problem in serious mathematic. In this topic, many essay were written. Here's my serious essay about "why earth ball", which has been rejected by arxiv and my mentors...... I would want to know if physicists are interest? I really think that is...
  47. L

    Show that the limit (1+z/n)^n=e^z holds

    Hi, I have problems proving task d I then started with task c and rewrote it as follows ##\lim_{n\to\infty}\sum\limits_{k=0}^{N}\Bigl( \frac{z^k}{k!} - \binom{n}{k} \frac{z^k}{n^k} \Bigr)=0 \quad \rightarrow \quad \lim_{n\to\infty}\sum\limits_{k=0}^{N} \frac{z^k}{k!} =...
  48. I

    Prove ##(a\cdot b)\cdot c =a\cdot (b \cdot c)## using Peano postulates

    with this background, we proceed to the proof. Let us define a set $$ G = \{ z \in \mathbb{N} | \; x, y \in \mathbb{N}\; (x \cdot y) \cdot z = x \cdot (y \cdot z) \} $$ We want to prove that ##G = \mathbb{N} ##. For this purpose, we will use part 3) of Peano postulates given above...
  49. I

    Prove ##a\cdot b = b \cdot a ##using Peano postulates

    with this background, we proceed to the proof. Let us define a set $$ G = \{ z \in \mathbb{N} | \mbox{ if } y \in \mathbb{N}, y\cdot z = z \cdot y \} $$ We want to prove that ##G = \mathbb{N} ##. For this purpose, we will use part 3) of Peano postulates given above. Obviously, ## G...
  50. I

    Prove ##(a+b)\cdot c=a\cdot c+b\cdot c## using Peano postulates

    I want to prove that ##(a+b)\cdot c=a\cdot c+b\cdot c## using Peano postulates where ##a,b,c \in \mathbb{N}##. The book I am using ("The real numbers and real analysis" by Ethan Bloch ) defines Peano postulates little differently. Following is a set of Peano postulates I am using. (Axiom 1.2.1...
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