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. Philethan

    A Bloch theorem proof with V(x)=V(x+ma)

    In Grosso's Solid State Physics, chapter 1, page 2, The author said that: Therefore, I plug (4) into (1), and I expect that I can get the following relationship, which proves that ##H\left|W_{k}(x)\right\rangle## belongs to the subspace ##\mathbf{S}_{k}## of plane waves of wavenumbers...
  2. J

    A Does the Frauchiger-Renner Theorem prove only MWI is correct

    Hello all, I have only seen this paper brought up here once before based on the search function 2 years ago, and the thread devolved into something off topic within the first page. I am asking in reference to this paper: https://arxiv.org/pdf/1604.07422.pdf Which claims to show that single...
  3. binbagsss

    Quick question on Laurent series proof uniqueness

    Homework Statement I am looking at the wikipedia proof of uniqueness of laurent series: https://en.wikipedia.org/wiki/Laurent_seriesHomework Equations look above or belowThe Attempt at a Solution I just don't know what the indentity used before the bottom line is, I've never seen it before...
  4. binbagsss

    Elliptic Functions Proof of Sum of Residues=0

    Homework Statement Hi I am looking at the attached proof for this property. I agree with the first line due to periodicity, but unsure about the next- see below 3)attempt Homework Equations To me, I deemed the integration substituion rule as relevant to this question, but perhaps...
  5. T

    I How can I go about making a "Space Proof" coating?

    So I am planning on launching a Satellite to promote the Dogecoin cryptocurrency. One of the main points is printing/painting (Or whatever) the logo on the side of a metal panel. How can I make it so it doesn't melt off or turn white from radiation so quickly?
  6. B

    Proof of oscillation about the equilibrium

    Homework Statement The problem is question 2(a) in the attached pdf. I seem to find myself at a dead end and am not sure where to go from here - I will attach my working in a separate file, but basically I need to show that the oscillator passes/crosses over the x = 0 boundary at a positive...
  7. Mr Davis 97

    I Understanding Proof of Uniqueness

    I'm trying to really get a grasp on proofs of uniqueness. Here is a model problem: Prove that ##x=-b/a## is the unique solution to ##ax+b=0##. First method: First we show existence of a solution: If ##x = -b/a##, then ##a(-b/a)+b = -b+b = 0##. Now, we show uniqueness: If ##ax+b=0##, then...
  8. Mr Davis 97

    I Understanding a Graph Theory Proof

    Prove that if a simple graph G has 6 vertices then G or its complement has a subgraph isomorphic to ##K_3##. The proof begins by noting that is must be the case that G or its complement as a vertex with degree at least 3. Why is this the case?
  9. isukatphysics69

    Not understanding calc proof of series

    Homework Statement Homework EquationsThe Attempt at a Solution I don't understand why for the first part where the series goes up until arn-1, it cannot just go up until arn.. why will that first series always go up until arn-1 until it is multiplied by r?
  10. Cheesycheese213

    B Fermat's little theorem proof?

    So I was taught that If gcd (a, p) = 1, then ap-1 ≡ 1 (mod p) And then the proof was Lemma: Let p be prime, Let i, j ,k = Integers If gcd (k, p) = 1 and ik ≡ jk (mod p) then i ≡ j (mod p) Main Proof: Consider 1a, 2a, 3a, ..., (p - 1)a Taking mod p is some arrangement of 1, 2, 3, ..., p - 1 Then...
  11. A

    I Proof that 1 is an odd number using Peano Axioms of naturals

    So I was just writing a proof that every natural number is either even or odd. I went in two directions and both require that 1 is odd, in fact I think that 1 must always be odd for every such proof as the nature of naturals is inductive from 1. I am using the version where 1 is the smallest...
  12. Mr Davis 97

    I Proof of Alternating Series Test

    I'm looking at the proof of the alternating series test, and the basic idea is that the odd and even partial sums converge to the same number, and that this implies that the series converges as a whole. What I don't understand is why the even and odd partial sums converging to the same limit...
  13. Mr Davis 97

    Epsilon-delta continuity proof

    Homework Statement Prove that ##f(x) = \frac{1}{x}## is continuous using the epsilon-delta definition of continuity. Homework EquationsThe Attempt at a Solution We will assume that the domain of ##f## is ##\mathbb{R} / \{ 0\}##. Let ##x_0## be in the domain. First, we look at ##\displaystyle...
  14. alexmahone

    MHB Could there be an error in the proof of the Poincare conjecture?

    When Grisha Perelman submitted his proof of the Poincare conjecture, he may have been reasonably sure that it contained no mistakes. But he could not have been 100% sure as he is, after all, human. Each time it was checked, say by the referee of an academic journal, the probability that it...
  15. cobalt124

    I Proof that BB(k) grows faster than any computable function

    Hi, layman post, not sure what thread level I need. From post https://www.physicsforums.com/threads/the-busy-beaver-function.942741/, I've been working my way through https://www.scottaaronson.com/busybeaver.pdf and come to section "1.3 The Busy Beaver Function", which states: "The Busy Beaver...
  16. M

    Logic Behind a Proof: Injective Function G

    Homework Statement suppose I have a function defined as: G: ℚ--->ℚ f(x)= { 2/ 3x if x does not equal to 0, 0 if x=0} Homework Equations Injective:if for all x,y in ℚ, f(x)=f(y) then x=y. or if x does not equal to y then f(x) does not equal to f(y)The Attempt at a Solution I am confused as to...
  17. rocdoc

    I Proof of Kaku (8.18): Completing the Square and Using Spiegel's Result

    In the following there is a proof, for positive values of ##a## only, of (8.18) of Kaku, reference 1, I quote' $$\int_{-\infty}^\infty~\mathrm{d}p~e^{iap^2+ibp}=\sqrt \frac{i\pi}{a}e^{-ib^2/4a}~~~~~~~~~~~~~(8.18)$$ '. Kaku says this result can be proved by completing the square. $$iap^2+ibp =...
  18. K

    I Proof of a law versus proof of a theorem

    if I get proof of fundamental laws like Newton's laws of motion or fundamental laws of thermodynamics then will they be laws anymore or will they become theorem. Please tell
  19. M

    Is this Proof of Equality Correct?

    Homework Statement Prove the following: If x=y and y=z then x=z. Now, this seems very obvious, and it is without a doubt correct. However, I am curious as to if the following proof is correct. Homework EquationsThe Attempt at a Solution Assume x does not equal to z, so that means two cases...
  20. Mr Davis 97

    I Proof that a sequence has two subsequential limits

    Suppose I have the sequence ##a_n = 2^{(-1)^n}##. So ##\displaystyle (a_n) = (\frac{1}{2},2,\frac{1}{2},2,\frac{1}{2},2,\frac{1}{2},2,...)##. Clearly, this sequence has two subsequential limits, ##\displaystyle \{\frac{1}{2},2 \}##. This clear from observation, but I'm not sure how I can be sure...
  21. Mr Davis 97

    I Proof that a quantity is greater than 1/2

    I'm looking at the quantity ##\displaystyle 1 - \frac{N}{n}##, and trying to prove that it is greater than ##1/2##, given that ##n> N##. I thought that since ##\lim_{n \to \infty} 1 - \frac{N}{n} = 1##, we could use the definition of convergence to get this inequality, for suitable ##\epsilon##...
  22. R

    When is the minimum polynomial of a scalar matrix kI equal to t-k?

    Homework Statement Show that A is a scalar matrix kI if and only if the minimum polynomial of A is m(t) = t-k Homework EquationsThe Attempt at a Solution f(A) is monic f(A) = 0 since A = kI Next we must show that deg(g) < deg(f) I guess I'm not sure where g comes from. Is it merely an...
  23. jamalkoiyess

    I Proof that p is interior if p is not limit of complement

    Hello PF, I am searching for a proof that I couldn't find on the internet. Theorem: E in X a metric space. p in E. p is an interior point of E if and only if p is not a limit point of (E complement)' Sorry for notations but I have no idea how to insert Latex here.
  24. Mr Davis 97

    I Clarification of a line in a proof

    This comes from a line of a proof in my book, and I need help resolving why the equality is true. Suppose that ##M>N##. Why is it true that ##\displaystyle \sup \{\frac{1}{n} (s_{N+1} + \cdots + s_n) ~|~ n>M \} = \frac{n-N}{n}\sup \{s_n ~|~ n > N \}##?
  25. Z

    Help with Newton root approximation proof

    Homework Statement Suppose we have: ## f(x) = x^2 - b ## ## b > 0 ## ## x_0 = b ## And a sequence is defined by: ## x_{i+1} = x_i - \frac{f(x_i)}{f'(x_i) } ## prove ## \forall i \in N ( x_i > 0 ) ## Homework Equations The Attempt at a Solution a)Here I tried solving for ## x_1 ## as...
  26. Physiona

    What is the algebraic proof for the remainder of 11 when dividing by 12?

    I'm currently doing a grade 9 paper, and one of the following questions is tripping me up a little bit: Prove algebraically that the sum of the squares of any three consecutive odd numbers always leaves a remainder of 11, when divided by 12. My attempt of the question: I have labelled 3...
  27. J

    I How Can Jensen's Inequality Be Used to Prove a Vector Magnitude Relationship?

    I have a vector B of length N, I would like to prove that: ∑n=0 to N-1 (|Bn|x) ≥ Nαx where: x > 1; α = (1/N) * ∑n=0 to N-1 (|Bn|) (i.e., The mean of the absolute elements of B). and ∑n=0 to N-1 (||Bn|-α|) ≠ 0; (i.e., The absolute elements of B are not all identical). I believe the above to...
  28. J

    MHB Polynomial Proof: Verification & Correction

    I would like to have verification if the following attached proof is correct. If it is not correct, what can be done to make it correct? Thanks.
  29. C

    MHB The proof of the infinite geometric sum

    Dear Everybody, I need some help with find M in the definition of the convergence for infinite series. The question ask, Prove that for $-1<r<1$, we have $\sum_{n=0}^{\infty} r^n=\frac{1}{1-r}$. Work: Let $\sum_{n=0}^{k} r^n=S_k$. Let $\varepsilon>0$, we must an $M\in\Bbb{N}$ such that $k\ge...
  30. barcodeIIIII

    Proof: Time independence of the entropy under unitary time evolution

    Homework Statement The unitary time evolution of the density operator is given by $$\rho(t)=\textrm{exp}(-\frac{i}{\hbar}Ht)\,\rho_0 \,\textrm{exp}(\frac{i}{\hbar}Ht)$$ General definition of entropy is $$S=-k_B\,Tr\,\{\rho(t) ln \rho(t)\}$$ Proof: $$\frac{dS}{dt}=0$$ Homework Equations I am not...
  31. O

    Is it possible to prove (P→Q)↔[(P ∨ Q)↔Q] without using truth tables?

    Homework Statement Need to demonstrate this proposition: (P→Q)↔[(P ∨ Q)↔Q] . My textbook use truth tables, but I'd like to do without it. It asks me if it's always truthThe Attempt at a Solution Im unable to demonstrate the Tautology and obtain (¬Q) as solution. I start by facing the right side...
  32. F

    Proving the Relationship between Lim Sup and Lim Inf

    Homework Statement Prove the ##\limsup \vert s_n \vert = 0## iff ##\lim s_n = 0##. Homework Equations ##\limsup s_n = \lim_{N\rightarrow \infty} \sup \lbrace s_n : n > N \rbrace = \sup \text{S}## ##\liminf s_n = \lim_{N\rightarrow \infty} \inf \lbrace s_n : n > N \rbrace = \inf \text{S}##...
  33. Mr Davis 97

    Real Analysis: Prove Upper Bound of Sum of Bounded Sequences

    Homework Statement Suppose that ##( s_n )## and ## (t_n)## are bounded sequences. Given that ##A_k## is an upper bound for ##\{s_n : n \ge k \}## and ##B_k## is an upper bound for ##\{t_n : n \ge k \}## and that ##A_k + B_k## is an upper bound for ##\{s_n + t_n : n \ge k \}##, show that ##\sup...
  34. Mr Davis 97

    Proof that a recursive sequence converges

    Homework Statement Prove that ##\displaystyle t_{n+1} = (1 - \frac{1}{4n^2}) t_n## where ##t_1=1## converges. Homework EquationsThe Attempt at a Solution First, we must prove that the sequence is bounded below. We will prove that it is bounded below by 0. ##t_1 = 1 \ge 0##, so the base case...
  35. mertcan

    Bland rule proof linear programming

    <Moderator's note: Continued from a technical forum and thus no template. Re-opening has been approved by moderator.> Hi, my question is related to simplex algorithm anticycling rule called Bland's rule. While I was working with the proof in the link...
  36. S

    B Did President Garfield really come up with an alternate proof?

    I'm talking about the Pythagorean Theorem, which seems to have an alternate proof attested to him! http://jwilson.coe.uga.edu/EMT668/EMT668.Student.Folders/HeadAngela/essay1/Pythagorean.html
  37. L

    MHB Question about proof of the linear independence of a dual basis

    This is from Kreyszig's Introductory Functional Analysis Theorem 2.9-1. Let $X$ be an n-dimensional vector space and $E=\{e_1, \cdots, e_n \}$ a basis for $X$. Then $F = \{f_1, \cdots, f_n\}$ given by (6) is a basis for the algebraic dual $X^*$ of $X$, and $\text{dim}X^* = \text{dim}X=n$...
  38. Stoney Pete

    I An easy proof of Gödel's first incompleteness theorem?

    Hi everybody, Do you think the following reconstruction of Gödel's first incompleteness theorem is basically correct, or at least in the right ballpark? In my view, this incompleteness result basically turns on the mismatch between the indenumerability of the powerset of ℕ and the enumerability...
  39. Eclair_de_XII

    Proof for convergent sequences, limits, and closed sets?

    Homework Statement "Let ##E \subset ℝ##. Prove that ##E## is closed if for each ##x_0##, there exists a sequence of ##x_n \in E## that converges to ##x_0##, it is true that ##x_0\in E##. In other words, prove that ##E## is closed if it contains every limit of sequences for each of its...
  40. RoboNerd

    Finding GCD with Fibonacci: Base Case

    Homework Statement Suppose that m divisions are required to find gcd(a,b). Prove by induction that for m >= 1, a >= F(m+2) and b>= F(m+1) where F(n) is the Fibonacci sequence. Hint: to find gcd(a,b), after the first division the algorithm computes gcd(b,r). Homework Equations Fibonacci...
  41. O

    Step potential, continuous wave function proof

    Homework Statement I am being asked to show that the wave function ψ and dψ/dx are continuous at every point of discontinuity for a step potential. I am asked to make use of the Heaviside step function in my proof, and to prove this explicitly and in detail. Homework Equations...
  42. C

    MHB Which fractions can you make (with proof)?

    Suppose you have the fraction 1/1. If you can make a fraction x/y, you can also make y/(2x). Also, if you can make x/y and a/b where GCD(x,y)=GCD(a,b)=1, you can make (x+a)/(y+b). Which fractions can you make?
  43. V

    B Fermat's Last Theorem; unacceptable proof, why?

    Wikipedia says Fermat's last theorem has the greatest number of failed proofs in history. I presume this simple "proof" is one of them. It must have been thought up before me. I first considered it years ago when I first heard of the problem. Figured it was so simple someone else must have...
  44. N

    An alternative proof (Hopefully not an alternative fact)

    Homework Statement Hi all, I'm currently studying the amazing Calculus by Spivak. Whenever I study textbooks I always attempt to do all the examples and proofs in the text before looking at the answers. (Whether this is a good thing or a bad thing I don't know, the examples are similar to the...
  45. Math Amateur

    MHB Proof of Bolzano-Weierstrass on R .... .... D&K Theorem 1.6.2 .... ....

    I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of the proof of Theorem 1.6.2 ... Duistermaat and Kolk"s Theorem 1.6.2 and its proof read as follows:In the...
  46. L

    Proof of uniqueness of limits for a sequence of real numbers

    Homework Statement [/B] The proposition that I intend to prove is the following. (From Terence Tao "Analysis I" 3rd ed., Proposition 6.1.7, p. 128). ##Proposition##. Let ##(a_n)^\infty_{n=m}## be a real sequence starting at some integer index m, and let ##l\neq l'## be two distinct real...
  47. Euler2718

    Can Three Non-Collinear Points Always Define a Projective Plane?

    Homework Statement Let P(W) be a projective space whose dimension is greater than or equal to 2 and let three non-colinear projective points, [v_{1}],[v_{2}],[v_{3}]\in P(W) . Prove that there is a projective plane in P(W) containing all three points. Homework EquationsThe Attempt at a...
  48. Mr Davis 97

    Proof by Contradiction: Showing a ≤ b when a ≤ b1 for every b1 > b

    Homework Statement Let ##a,b \in \mathbb{R}##. Show if ##a \le b_1## for every ##b_1 > b##, then ##a \le b##. Homework EquationsThe Attempt at a Solution We will proceed by contradiction. Suppose that ##a \le b_1## for every ##b_1 > b##, and ##a > b##. Let ##b_1 = \frac{a+b}{2}##. We see that...
  49. C

    A Is the proof of these results correct?

    Hello, Below are two results with their proof. Of course, there may be several ways to prove these results, but I just need some checking. Can someone check carefully if the math is OK ? (but very carefully, because if there is a failure, I will be murdered :-) ) ? thx. Claim 1: Let ##L/K## be...
  50. J

    MHB Congruence Class Proofs: Tips and Examples

    Hi, I have tried the question as attached. I am not sure if I am correct. You help is greatly appreciated. Thanks in advance!
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