What is Proof: Definition and 999 Discussions

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. T

    Experimental proof of Venturi Effect

    Venturi effect is known for centuries. And most probably that's why experimental proofs are rare because it's already accepted. But, I want to know how close real results are in case of experiments regarding Venturi Effect. I am especially interested in results of experiment regarding velocity...
  2. A

    I Proving the Finite Binomial Series for k Non-Negative Integer

    Hello, I was wondering how to prove that the Binomial Series is not infinite when k is a non-negative integer. I really don't understand how we can prove this. Do you have any examples that can show that there is a finite number when k is a non-negative integer? Thank you!
  3. Samkiwi

    How is the Relativity Formula Proven for an Electron in an Electric Field?

    It is an electron initially pushed by the action of the electric field. The vectors of force and velocity are parallel to each other. Here's the questionA possible expression of speed as a function of time is the following: $$v(t) = \frac{At}{\sqrt{1 + (\frac{At}{c})^2}}$$where is it $$A...
  4. Samkiwi

    Deriving and Verifying the Relativity Formula for Electron Speed

    It is an electron initially pushed by the action of the electric field. The vectors of force and velocity are parallel to each other. Here's the questionA possible expression of speed as a function of time is the following: $$v(t) = \frac{At}{\sqrt{1 + (\frac{At}{c})^2}}$$where is it $$A...
  5. Eclair_de_XII

    B Is this a valid proof for the Extreme Value Theorem?

    If ##f## is a constant function, then choose any point ##x_0##. For any ##x\in K##, ##f(x_0)\geq f(x)## and there is a point ##x_0\in K## s.t. ##f(x_0)=\sup f(K)=\sup\{f(x_0)\}=f(x_0)##. Now assume that ##f## is not a constant function. Construct a sequence of points ##x_n\in K## as follows...
  6. Z

    Proving Energy Conservation in a Gravitational System with Multiple Bodies

    Hi all. I'm trying to prove energy conservation in a (maybe) uncommon way. I know there are different ways to do this, but it is asked me to prove it this way and I'm stucked at the end of the proof. I'm considering ##N## bodies moving in a gravitational potential, such that the energy is ##E =...
  7. C

    I Proof check for Plancherel's theorem (Fourier Transform version)

    I'm trying to prove Plancherel's theorem for functions $$f\in L^1\cap L^2(\mathbb{R})$$. I've included below my attempt and I would really appreciate it if someone could check this for me please, and give me any feedback they might have. **Note:** I am working with a slightly different...
  8. B

    Do Cauchy Sequences Imply Convergent Differences?

    I've started by writing down the definitions, so we have $$x_n-y_n\rightarrow 0\, \Rightarrow \, \forall w>0, \exists \, n_w\in\mathbb{N}:n>n_{w}\,\Rightarrow\,|x_n-y_n|<w $$ $$(x_n)\, \text{is Cauchy} \, \Rightarrow \,\forall w>0, \exists \, n_0\in\mathbb{N}:m,n>n_{0}\,\Rightarrow\,|x_m-x_n|<w...
  9. N

    MHB What is the Epsilon-Delta Method for Proving Limits?

    Use the epsilon-delta method to show that the limit is 3/2 for the given function. lim (1 + 2x)/(3 - x) = 3/2 x-->1 I want to find a delta so that | x - 1| < delta implies |f(x) - L| < epsilon. | (1 + 2x)/(3 - x) - (3/2) | < epsilon -epsilon < (1 + 2x)/(3 - x) - 3/2 < epsilon I now add...
  10. Frabjous

    I Probability fun time: Proof that 1/3=1/2=1/4

    Forgive me, I am not a probability guy, so I am unsure how well known this is. I was trying to figure something out and found this. I found it cool. Here's the explanation. The first solution is a fraction (damn scanner!) Oops! From Kendall Geometrical Probability (1963)
  11. M

    MHB Logic Proof With Rules of Replacement

    Not sure if this is an allowed post, as it is not technically math but I'm trying to work through the below proof. If workers have a fundamental right to a job, then unemployment will be virtually nonexistent but job redundancy will become a problem. If workers have no fundamental right to a...
  12. larginal

    Proof of Electromagnetic Identity: Puzzling Last Expression

    I tried to understand proof of this identity from electromagnetics. but I was puzzled at the last expression. why is that line integral of dV = 0 ? In fact, I'm wondering if this expression makes sense.
  13. mcastillo356

    B Proof of Chain Rule: Understanding the Limits

    First I quote the text, and then the attempts to solve the doubts: "Proof of the Chain Rule Be ##f## a differentiable function at the point ##u=g(x)##, with ##g## a differentiable function at ##x##. Be the function ##E(k)## described this way: $$E(0)=0$$...
  14. C

    Indirect Proof Proof verification: sequence a_n=(−1)^n does not converge

    Theorem: Show that the sequence ## a_n = (-1)^n ## for all ## n \in \mathbb{N}, ## does not converge. My Proof: Suppose that there exists a limit ##L## such that ## a_n \rightarrow L ##. Specifically, for ## \epsilon = 1 ## there exists ## n_0 ## s.t. for all ## n > n_0## then ##|(-1)^n-L|<1##...
  15. S

    MHB Need help with parallelogram proof

    Hello, we are learning about similar triangles and this was a problem. So I know that opposite sides of a parallelogram are congruent as are opposite angles, so I can establish similarity with triangles WYS and STW, but I don't understand how that proves SX x YW = SV x WT because the proportions...
  16. C

    I How to deal with self-doubt in mathematics?

    Dear Everyone, I am wondering how to deal with the self-doubt in proof-writing in general situation like on exam or homework question. Suppose I want to prove Theorem B. I assume the hypothesis. Then I apply the right mathematics definition. I am hesitant on the next step; I have the feeling...
  17. Demystifier

    A A stronger proof of nonlocality, or what?

    Half a year ago a group of authors published a paper in Nature Physics https://www.nature.com/articles/s41567-020-0990-x which seems to be a proof of nonlocality even stronger than Bell nonlocality. More precisely, according to a popular exposition by one of the authors...
  18. L

    MHB How Can the 4 Rod Tower of Hanoi Inequality Be Solved?

    I am having trouble solving part 2, for $ W_{\frac{n(n+1)}{2}} \leq 2^{n} (n-1) + 1 , n \geq 0 $ I know that $W_{m} \leq 2*W_{m-k} + 2^{k} – 1, 0 \leq k \leq m$ Let $m = \frac{n(n+1)}{2}$ So now $W_{\frac{n(n+1)}{2}} \leq 2*W_{\frac{n(n+1)}{2} - k} + 2^{k} - 1, 0 \leq k \leq...
  19. C

    Existence of isomorphism ϕ:V→V s.t. ϕ(ϕ(v))=−v for all v∈V

    Problem: Let ## V ## be a vector space over ## \mathbb{F} ## and suppose its dimension is even, ## dimV=2k ##. Show there exists an isomorphism ## \phi:V→V ## s.t. ## \phi(\phi(v))=−v ## for all ## v \in V ## Generally that way to solve this is to define a basis for the vector space ## V ##...
  20. G

    B Light Speed Invariance: Experiments, Difficulties & Clarification

    Let me clarify my question, is there any experiment directly proved the invariance of light speed to observers? Let's not get to the argument of equivalence between source and observer. SR was based on the postulate that the light speed is constant and independent of both the motions of source...
  21. cybernetichero

    What WOULD be adequate proof of alien visitation to Earth?

    Perseverance's successful landing has scared up some UFO conspiracists online. I used to be like them and I was into Charles Forte as well, encouraged by people who should have known better (yes I AM looking at you Arthur C. Clarke) until I realized I just really wanted to believe rather than...
  22. D

    Inequality proof: If a>b implies a>c then b>c

    Summary:: To prove a conditional statement on a pair of inequalitites. Mentor note: Moved from technical forum section, so the post is missing the usual fields. I feel it should be possible to prove this but I keep getting lost in the symbolic manipulation. Theorem: If a>b implies a>c then...
  23. Armine

    Proof of a formula with two geometric random variables

    The image above is the problem and the image below is the solution I have tried but failed.
  24. S

    MHB Proof of Triangle Inequality for $n$ Natural Numbers

    Prove for all $n\in N$ $\dfrac{|a_1+...a_n|}{1+|a_1+...+a_n|}\leq\dfrac{|a_1|}{1+|a_1|}+...\dfrac{|a_n|}{1+|a_n|}$
  25. C

    Det of Triangular-like Matrix & getting stuck in Algorithmic Proof

    Find determinant of following matrix: ## A = \begin{pmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,n-1} & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,n-1} & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{n,1} & 0 & \cdots & 0 & 0 \end{pmatrix} ## Note: I tried to solve this question...
  26. DaTario

    I Proof for a congruence relation

    Hi, I would like to prove the following congruence relation: Let p be a prime number and let ##n## be a natural such that ##p < n < p^2##. Then $$ {n-1 \choose p-1} {n \choose p-1} \equiv 0 (\mbox{mod p}) .$$ I am expecting it to have a rather trivial proof. Thanks in advance for any...
  27. D

    I Help With a Proof using Contour Integration

    I am reading a proof in Feedback Systems by Astrom, for the Bode Sensitivity Integral, pg 339. I am stuck on a specific part of the proof. He is evaluating an integral along a contour which makes up the imaginary axis. He has the following: $$ -i\int_{-iR}^{iR}...
  28. Eclair_de_XII

    Am I using quotient spaces correctly in this linear algebra proof?

    %%% Assume that ##X/Y## is defined. Since ##\dim Y = \dim X##, it follows that ##\dim {X/Y}=0## and that ##X/Y=\{0\}##. Suppose that ##Y## is a proper subspace of ##X##. Then there is an ##x\in X## such that ##x\notin Y##. Let us consider the equivalence class: ##\{x\}_Y=\{x_0\in...
  29. J

    Confused with this proof for the Cauchy Schwarz inequality

    Im confused as finding the minimum value of lambda is an important part of the proof but it isn't clear to me that the critical point is a minimum
  30. P

    I Proof concerning the Four Fundamental Spaces

    Hello all, I am currently working on the four fundamental spaces of a matrix. I have a question about the orthogonality of the row space to the null space column space to the left null space ------------------------------------------------ In the book of G. Strang there is this nice picture...
  31. J

    I Spacetime invariance algebraic proof

    In Phillip Harris' (U. Sussex) post on special relativity he includes on p. 45 an algebraic proof of invariance of spacetime intervals. He starts with the definition S^2 =c^t^2 - x^2 -y^2 -z^2, he inserts the Lorentz transform expressions fot t and x, and he does some algebra to show that one...
  32. yucheng

    Can I use recursion/induction to show that N <= x < N+1 for x real

    Homework Statement:: Show that for every real number ##x## there is exactly one integer ##N## such that ##N \leq x < N+1##. (This integer is called the integer part of ##x##, and is sometimes denoted ##N = \lfloor x\rfloor##.) Relevant Equations:: N/A I have tried reading the solution given...
  33. yucheng

    Is my proof that multiplication is well-defined for reals correct?

    I have referred to this page: https://taoanalysis.wordpress.com/2020/03/26/exercise-5-3-2/ to check my answer. The way I thought of the problem: I know ##xy = \mathrm{LIM}_{n\to\infty} a_n b_n## and I know ##x'y = \mathrm{LIM}_{n\to\infty} a'_n b_n##. Thus if ##xy=x'y##, maybe I can try showing...
  34. yucheng

    Understanding the Use of Min in Cauchy Sequences

    I refer to this page: https://taoanalysis.wordpress.com/2020/03/26/exercise-5-3-2/ I am having trouble understanding the purpose / motivation behind using the min as in ##\delta := \min\left(\frac{\varepsilon}{3M_1}, 1\right)## and ##\varepsilon' := \min\left(\frac{\varepsilon}{3M_2}...
  35. yucheng

    Contradictory Proof of Supremum of E: Is it Circular?

    N.B. I have inserted the proof here as reference. See the bolded text. My question is, isn't the reasoning "##x^{2}+5 \varepsilon<2,## thus ##(x+\varepsilon)^{2}<2 .## " circular? If we can already find an ##0<\varepsilon<1## such that ##x^{2}+5 \varepsilon<2,## Can't one also claim that " we...
  36. yucheng

    I Is ##\delta##-steady needed in this proof, given ##\epsilon##-steady

    In Tao's Analysis 1, Lemma 5.3.6, he claims that "We know that ##(a_n)_{n=1}^{\infty}## is eventually ##\delta##-steady for everyvalue of ##\delta>0##. This implies that it is not only ##\epsilon##-steady, ##\forall\epsilon>0##, but also ##\epsilon/ 2##-steady." My question is, why do we need...
  37. yucheng

    Proof that two equivalent sequences are both Cauchy sequences

    Let us just lay down some definitions. Both sequences are equivalent iff for each ##\epsilon>0## , there exists an N>0 such that for all n>N, ##|a_n-b_n|<\epsilon##. A sequence is a Cauchy sequence iff ##\forall\epsilon>0:(\exists N>0: (\forall j,k>N:|a_j-a_k|>\epsilon))##. We proceeded by...
  38. M

    Integral proof and then use the proof to solve a 2nd integral

  39. R

    If P = NP, can a proof be generated / verified efficiently for a proposition?

    If P = NP, then the solution of any decision problem in NP can be found efficiently. Consider the Pythagoras theorem. It can be represented as an encoded (binary) string. (How?) Let D be the decision problem asking whether the input to D is a proof of the Pythagoras theorem. Given an...
  40. Andrei0408

    Eigenfunction proof and eigenvalue

    I searched through the courses but I can't find any formula to help me prove that the expression is an eigenfunction. Am I missing something? What are the formulas needed for this problem statement?
  41. patric44

    Proof related to the center of mass

    hi guys in the proof of the parallel axis theorem this equation is just put as it is as a definition of the center of mass : $$\int[2(\vec{r_{o}}.\vec{r'})I-(\vec{r_{o}}⊗\vec{r'}+\vec{r'}⊗\vec{r_{0}})]dm = 0$$ is there is any proof for this definition ? and what is the approach for it
  42. L Navarro H

    Proof that the exponential function is convex

    I try to proof it but i got stuck right here, i want your opinions Can I get a solution if i continue by this way? or Do I have to take another? and if it is so, what would yo do?
  43. C

    I Supremum proof & relation to Universal quantifier

    In the following proof: I didn't understand the following part: Isn't it supposed to be : ## a > s_A - \epsilon >0 ## and ## b > s_B - \epsilon >0 ## Because to prove that ## s ## is a supremum, we need to prove the following: For every ## \epsilon > 0 ## there exists ## m \in M ## such...
  44. L

    MHB Is G/G isomorphic to the trivial group? A proof for G/G\cong \{e\}

    Reorder the statements below to give a proof for G/G\cong \{e\}, where \{e\} is the trivial group. The 3 sentences are: For the subgroup G of G, G is the unique left coset of G in G. Therefore we have G/G=\{G\} and, since G\lhd G, the quotient group has order |G/G|=1. Let \phi:G/G\to \{e\} be...
  45. C

    Simple Induction Help with Lemma for proof of AM-GM inequality

    Summary:: x Hey, I'm learning calculus and had to prove the following Lemma which is used to prove AM-GM inequality, I had tried to prove it on my own and it is quite different from what is written in my lecture notes. I have a feeling that my proof of the Lemma is incorrect, but I just don't...
  46. E

    B Don't understand proof of Bloch theorem

    The potential inside the crystal is periodic ##U(\vec{r} + \vec{R}) = U(\vec{r})## for lattice vectors ##\vec{R} = n_i \vec{a}_i##, ##n_i \in \mathbb{Z}## (where the ##\vec{a}_i## are the crystal basis), and Hamiltonian for an electron in the crystal is ##\hat{H} = \left( -\frac{\hbar^2}{2m}...
  47. patric44

    Proof of the generalized Uncertainty Principle?

    hi guys i am trying to follow a proof of the generalized uncertainty principle and i am stuck at the last step : i am not sure why he put these relations in (4.20) : $$(\Delta\;C)^{2} = \bra{\psi}A^{2}\ket{\psi}$$ $$(\Delta\;D)^{2} = \bra{\psi}B^{2}\ket{\psi}$$ i tried to prove these using the...
  48. K

    Proof question related to the Ideal Gas Law

    A cylinder contains an initial volume V1 = 1m^^3 of a perfect gas at initial pressure p1 = 1 bar, confined by a piston that is held in place by a spring. The gas is heated until its volume is doubled and the final pressure is 5 bar. Assuming that the mass of the piston is negligible and that the...
  49. chocopanda

    Harmonic oscillator with ladder operators - proof using the Sum Rule

    I'm trying verify the proof of the sum rule for the one-dimensional harmonic oscillator: $$\sum_l^\infty (E_l-E_n)\ | \langle l \ |p| \ n \rangle |^2 = \frac {mh^2w^2}{2} $$ The exercise explicitly says to use laddle operators and to express $p$ with $$b=\sqrt{\frac {mw}{2 \hbar}}-\frac...
  50. B

    I Proving Probability of Union with Indicator Variables in Three Events

    "Prove Theorem 7.1 about the probability of a union, using the 12.3 proof (see section 12.2) that involves indicator variables. Do not write the proof in full generality, only for three events. You should not use the product notation; you should write out all factors of the product." I'm taking...
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