Proof Definition and 999 Threads

I am struggling to understand the induction proof of the pigeonhole principle in my textbook. The theorem and the proof, from Biggs Discrete Mathematics, is pasted below, and I will explain further (see bold text) what I am having trouble with. Theorem. Let m be a natural number. Then the...

Proposition(Zorn's Lemma): Let ##X\neq\emptyset## be of partial order with the property that ##\forall Y\subseteq X## such that ##Y## is of total-order then ##Y## has an upperbound, then ##X## contains a maximal element. Proof: Case 1: ##B\neq\emptyset## such that ##B##=##\{####b\in X##: ##b##...

Hi since U.S. education is shite, I've decided that I'm going to learn math from the ground up by myself. My goal is to reach graduate level mathematics in 2-3 years. I'm currently reading Book of Proof, what should I read after this? My end goal is to be proficient in applied math/ physics.

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...

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...

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...

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?

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...

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...

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?

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?

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...

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...

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...

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...

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...

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...

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...

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 =...

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

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...

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...

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##...

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...

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.

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 \}##?

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...

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...

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...

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.

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...

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...

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...

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}##...

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...

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...

<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...

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

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 $\text{dim}X^* = \text{dim}X=n$...

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...

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...

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...

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...

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?

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...

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...

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...

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...

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...

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...