Proof Definition and 999 Threads

Prove that if a subsequence of a Cauchy sequence converges then so does the original Cauchy sequence. I'm assuming that we're not allowed to use the fact that every Cauchy sequence converges. Here's my attempt: Let $\displaystyle\{s_n\}$ be the original Cauchy sequence. Let $\displaystyle...

Theorem: Let ##A_1, A_2, ..., A_k## be finite, disjunct sets. Then ##|A_1 \cup A_2 \cup \dots \cup A_k| = |A_1| + |A_2| + \dots + |A_k|## I will give the proof my book provides, I don't understand several parts of it. Proof: We have bijections ##f_i: [n_i] \rightarrow A_i## for ##i \in [k]##...

Hello. Given a range of time in which an event can occur an indefinite number of times, we say a random variable X folows a poisson distribution when it follows this statements: X is the number of times an event occurs in an interval and X can take values 0, 1, 2, … The occurrence of one event...

Homework Statement We have JUST started writing proofs recently, and I am a little bit doubtful in my abilities in doing this, so I just want to verify that my proof actually works. I was expecting this one to be a lot longer since the previous 2 were. I don't see any glaring flaws in it, but...

Hello all. I have a question concerning following proof, Lemma 1. http://planetmath.org/allbasesforavectorspacehavethesamecardinalitySo, we suppose that A and B are finite and then we construct a new basis ##B_1## for V by removing an element. So they choose ##a_1 \in A## and add it to...

From drchinese website http://www.drchinese.com/David/Bell_Compact.pdf on page 406 there are 2 equations at the top. How do you get from the top one to the second one? There is a hint about using (1) but I think it cannot be done. You might be able to do it with other assumptions but I think...

Homework Statement Show that the launch angle θ is given by the expression: θ=tan-1(4hmax/R) where hmax is the maximum height in the trajectory and R is the range of the projectile. Homework Equations hmax=vi2sin2(θ)/2g R=vi2sin(2θ)/g The Attempt at a Solution I am trying to understand the...

Homework Statement Prove that ##A \cup (A \cap B) = A## Homework Equations In the previous exercise, we proved: Let A, B be sets. Then, the following statements are equivalent: 1) ##A \subseteq B## 2) ##A \cup B = B## 3) ##A \cap B = A## The Attempt at a Solution The proof of ##A \cup (A...

A particle's position is given by $$r_i=r_i(q_1,q_2,...,q_n,t)$$ So velocity: $$v_i=\frac{dr_i}{dt} = \sum_k \frac{\partial r_i}{\partial q_k}\dot q_k + \frac{\partial r_i}{\partial t} $$ In my book it's given $$\frac{\partial v_i}{\partial \dot q_k} = \frac{\partial r_i}{\partial q_k}$$...

Homework Statement Prove that ##f: \mathbb{R}\to\mathbb{R}, f(x) = x^2## is not injective. Homework Equations Definition of an injection: function ##f:A\to B## is an injection if and only if ##\forall a,b \in A, f(a) = f(b) \Rightarrow a = b##. The Attempt at a Solution ##f...

I'm using the book of Jerome Keisler: Elementary calculus an infinitesimal approach. I have trouble understanding the proof of the following theorem. I'm not sure what it means. Theorem: "An increasing sequence <Sn> either converges or diverges to infinity." Proof: Let T be the set of all real...

1. I have to show: 2. Given: 3. My attempt : I just want to verify if what I've done is correct or not. Thanks!

I need to understand something about proof by contradiction. Suppose there is an expression "a" and it is known to be equal to expression "b". Furthermore, suppose it is conjectured that expression "c" is also equal to expression "a". This would imply expression "c" is equal to expression...

I have purchased an article after recommendation on wikipedia that as far as I am aware proves eotvos law. Here is a quote from wikipedia from this site: https://en.wikipedia.org/wiki/Eötvös_rule: ''John Lennard-Jones and Corner published (1940) a derivation of the equation by means of...

Homework Statement Show that n!>2n for all n>3. Homework Equations I will attempt to use induction. The Attempt at a Solution We want to show that n!>2n for all n>3. Consider the case when n=4. 4! = 24 > 2^4 =16. We want to show by way of induction that if the inequality is true for...

Homework Statement Use the definition of exclusive or (XOR), the facts that XOR commutes and associates (if you need this) and all the non-XOR axioms and theorems you know from Boolean algebra to prove this distributive rule: A*(B (XOR) C) = (A*B) (XOR) (A*C) Homework Equations All the...

This is from a general relativity book but I think this is the appropriate location. The proof that \nabla_{[a} {R_{bc]d}}^e=0 is as follows: Choose coordinates such that \Gamma^a_{bc}=0 at an event. We have \nabla_a {R_{bcd}}^e = \partial_a \partial_b \Gamma^e_{cd} - \partial_a...

Homework Statement $$ \left | \frac{z}{\left | z \right |} + \frac{w}{\left | w \right |} \right |\left ( \left | z \right | +\left | w \right | \right )\leq 2\left | z+w \right | $$ Where z and w are complex numbers not equal to zero. 2.$$\frac{z}{\left | z \right | ^{2}}=\frac{1}{\bar{z}}$$...

I have read a proof but I have a question. To give some context, I first wrote down this proof as written in the book. First, I provide the recursion theorem though. Recursion theorem: Let H be a set. Let ##e \in H##. Let ##k: \mathbb{N} \rightarrow H## be a function. Then there exists a...

I am stuck with one proof and I need some help because I don't have any idea how to proceed at this moment. The task says: If f(x) is a polynomial with integer coefficients, and if f(a)=f(b)=f(c)=-1, where a,b,c are three unequal integers, the equation f(x)=0 does not have integer solutions...

Recently I can across this proof: 0/0= (100-100)/(100-100) = (10^2-10^2)/10(10-10) =(10-10)(10+10)/10(10-10) = (10+10)/10 =20/10 = 2 But this is obviously wrong, as 0/0 is infinity, but which line in this proof is actually wrong?

Let's have waves with their carried power proportional to the square of their amplitude(s). The waves obey the principle of superposition. Before superposition, we can calculate the power output based on the amplitudes. After superposition, there will be new values for the amplitudes, but the...

1. Homework Statement Proof Let F:A→B and g: B→C be functions. Suppose that g°f is injective. Prove that f is injective. Homework EquationsThe Attempt at a Solution Let x,y ∈ A, and suppose g°f (x) = g°f(y) and x = y. Suppose g: B→C was not injective. then f(g(x)), if g(x) is some element...

Hello. I'm currently studying about natural numbers and I encountered the theorem of definition by recursion: This states: Let ##H## be a set, let ##e \in H## and let ##k: H \rightarrow H## be a function. Then there is a unique function ##f: \mathbb{N} \rightarrow H## such that ##f(1) = e##...

1. Homework Statement Derive the result DeltaT <0 for U> (sqrt(1-v^2/c^2)+1)/v/c)c Homework Equations DeltaT = u/l + (u-v/1-uv/c^2)1/l Where: DeltaT is the time for the tachyon to go and come back. u is the velocity of the tachyon l is the distance that the tachyon goes v is the velocity...

Homework Statement I'm asked to prove that Et - p⋅r = E't' - p'⋅r' Homework Equations t = γ (t' + ux') x = γ (x' + ut') y = y' z = z' E = γ (E' + up'x) px = γ (p'x + uE') py = p'y pz = p'z The Attempt at a Solution Im still trying to figure out 4 vectors. I get close to the solution but I...

In Spivak's Calculus, on page 121 there is this theorem Then he generalizes that theorem: I tried proving theorem 4 on my own, before looking at Spivak's proof. Thus I let c = 0 and then by theorem 1, my proof would be completed. Is this a correct proof? Spivak's proof for theorem 4...

Hi, One silly thing is bothering me. As per one lemma, If a, b, and c are positive integers such that gcd(a, b) = 1 and a | bc, then a | c. This is intuitively obvious. i.e. Since GCD is 1 'a' does not divide 'b'. Now, 'a' divides 'bc' so, 'a' divides 'c'. Proved. What is bothering me is ...

Homework Statement Suppose r:R\rightarrow { V }_{ 3 } is a twice-differentiable curve with central acceleration, that is, \ddot { r } is parallel with r. a. Prove N=r\times \dot { r } is constant b. Assuming N\neq 0, prove that r lies in the plane through the origin with normal N. Homework...

Hello everyone! I want to proof that: ##p \land (p \to q) \Rightarrow q## I know this is a quite trivial problem using truth tables, however, I want to do it without it. As I'm learning this myself, is this the correct approach? ##p \land (p \to q)## ##\iff p\land (\neg p \lor q)## ##\iff (p...

just starting out with proofs... i tried manipulating the right side of the inequality, but i don't see why it's equal to (n+1)!

The Rearrangement Inequality states that for two sequences ##{a_i}## and ##{b_i}##, the sum ##S_n = \sum_{i=1}^n a_ib_i## is maximized if ##a_i## and ##b_i## are similarly arranged. That is, big numbers are paired with big numbers and small numbers are paired with small numbers. The question...

Homework Statement Let P be a point on the sphere with center O, the origin, diameter AB, and radius r. Prove the triangle APB is a right triangle Homework Equations |AB|^2 = |AP|^2 + |PB|^2 |AB}^2 = 4r^2 The Attempt at a Solution Not sure if showing the above equations are true is the...

Homework Statement Let ##A## be an n × p matrix and ##B## be an p × m matrix with the following column vector representation, B = \begin{bmatrix} b_1 , & b_2, & ... & ,b_m \end{bmatrix} Prove that AB = \begin{bmatrix} Ab_1 , & Ab_2, & ... & , Ab_m \end{bmatrix} If ##A## is represented...

Homework Statement Hi, Just watching Susskind's quantum mechanics lecture notes, I have a couple of questions from his third lecture: Homework Equations [/B] 1) At 25:20 he says that ## <A|\hat{H}|A>=<A|\hat{H}|A>^*## [1] ##<=>## ##<B|\hat{H}|A>=<A|\hat{H}|B>^*=## [2] where ##A## and ##B##...

Homework Statement With ##\vec{r}## the position vector and ##r## its norm, we define $$ \vec{f} = \frac{\vec{r}}{r^n}.$$ Show that $$ \nabla^2\vec{f} = n(n-3)\frac{\vec{r}}{r^{n+2}}.$$ Homework Equations Basic rules of calculus. The Attempt at a Solution From the definition of...

A spherical snowball is melting at a rate proportional to its surface area. That is, the rate at which its volume is decreasing at any instant is proportional to its surface area at that instant. (i) Prove that the radius of the snowball is decreasing at a constant rate. can someone help me?

This is a linear algebra question which I am confused. 1. Homework Statement Prove that "if the union of two subspaces of ##V## is a subspace of ##V##, then one of the subspaces is contained in the other". The Attempt at a Solution Suppose ##U##, ##W## are subspaces of ##V##. ##U \cup W##...

Proof by contradiction starts by supposing a statement, and then shows the contradiction. 1. Homework Statement Now, there is a statement ##A##. Suppose ##A## is false. It leads to contradiction. So ##A## is true. My question: There are two statements ##A## and ##B##. Suppose ##A## is true...

1. Homework Statement prove the following statement: Hello, can someone help me prove this statement A is hermitian and {|Ψi>} is a full set of functions Homework Equations Σ<r|A|s> <s|B|c>[/B]The Attempt at a Solution Since the right term of the equation reminds of the standard deviation, I...

1. The derivation In a 3-dim space,a particle is acted by a central force(the center of the force fixed in the origin) .we now take the motion entirely in the xy-plane and write the equations of the motion in polar coordinate how can i derive from these equation that T(kinetic...

The proofs of the Fundamental Theorem of Calculus in the textbook I'm reading and those that I have found online, basically show us: 1) That when we apply the definition of the derivative to the integral of f (say F) below, we get f back. F(x) = \int_a^x f(t) dt 2) That any definite integral...

Homework Statement show that the general solution of the differential equation d^2/dt^2 + 2 *alpha * dr/dt + omega^2 * r = 0, where alpha and w are constant and R is a function of time "t" is R = e^(-alpha * t) * [ C1*sin( sqrt(omega^2 - alpha^2) * t) + C2*cos( sqrt(omega^2 - alpha^2) * t)...

I don't see how it proves that (n-r+1, r) is the number of r-combinations of X which contain no consecutive integers.

I am usually pretty good about interpreting what a question is asking when it is in the form, "prove that if p, then q," where p and q are statements. However, I cannot seem to understand how to interpret when it is in the form "prove that p if and only if q." The statement I am working with...

Homework Statement Let n>=2 n is natural and set x=(1,2,3,...,n) and y=(1,2). Show that Sym(n)=<x,y> Homework EquationsThe Attempt at a Solution Approach: Induction Proof: Base case n=2 x=(1,2) y=(1,2) Sym(2)={Id,(1,2)} (1,2)=x and Id=xy so base case holds Inductive step assume...

Homework Statement Eliminate t from the equation (x-xi)=vi(t)+1/2(a)t^2 using the kinematic equation v=vi+at to get v^2=vi^2+2a(x-xi) The Attempt at a Solution I wind up with (x-xi)=vi(v-vi/a) + 1/2(v^2-vi^2/a). If the first term on the right side didn't exist, I could see what the solution...

So I am working on this simple proof, but am confused about the term "external angle." The problem says that if ##a##, ##b##, and ##c## are external angles to a triangle, then ##a + b + c = 360##. However, is seems that the vertex of each triangle has two possible external angles, since there...

The problem statement: Show that if ##r_1## and ##r_2## are the distinct real roots of ##x^2 + px + 8 = 0##, then ##r_1 + r_2 > 4 \sqrt{2}##. We start by noting that ##r_1 r_2 = 8##. Using this relation, we'll find the minimum value of ##r_1 + r_2##. To minimize ##r_1 + r_2##, we need to...

This is from text [Introduction to Lagrangian and Hamiltonian Mechanics] on NTNU opencourse. Annnnd... I don't use english as my primary language, so sorry for poor sentences. I can't get two things in here. First, at (1.12) I can't understand how L dot derivated like that. Since I know...