Bounded Definition and 514 Threads

Homework Statement "Let ##\{a_n\}_{n=1}^\infty## be a bounded, non-monotonic sequence of real numbers. Prove that it contains a convergent subsequence." Homework Equations Monotone: "A sequence ##\{\alpha_n\}_{n=1}^\infty## is monotone if it is increasing or decreasing. In other words, if a...

Can (0,1)\subset\mathbb{R} be divided into an infinite set S of non-empty disjoint subsets? It seams like any pair of points in different subsets of the partitioning must have a finite difference, and so there must be some smallest finite difference overall, d where |S| \leq 1/d. Can someone...

Hi thanks to George, I found the following criteria for boundedness: \begin{equation} \frac{||Bf(x)||}{||f(x)||} < ||Bf(x)|| \end{equation} If one takes f(x) = x, and consider B = (h/id/dx - g), where g is some constant, then B is bounded in the interval 0-##\pi##. However, given that I...

Hi, I have an operator which does not obey the following condition for boundedness: \begin{equation*} ||H\ x|| \leqslant c||x||\ \ \ \ \ \ \ \ c \in \mathscr{D} \end{equation*} where c is a real number in the Domain D of the operator H. However, this operator is also not really unbounded...

Hey! :o Let $r_1,r_2,r_3, \ldots$ a numeration of all rational numbers and $f:\mathbb{R}\rightarrow \mathbb{R}$ with $\displaystyle{f(x)=\sum_{r_n<x}2^{-n}}$ I want to show that $f$ is bounded and strictly increasing. To show that the function is bounded, do we use the geometric sum...

If we were to quantize the Dirac field using commutation relations instead of anticommutation relations we would end up with the Hamiltonian, see Peskin and Schroeder $$ H = \int\frac{d^3p}{(2\pi)^3}E_p \sum_{s=1}^2 \Big( a^{s\dagger}_\textbf{p}a^s_\textbf{p}...

I am reading John B. Conway's book, "Functions of a Complex Variable I" (Second Edition) ... I am currently focussed on Chapter IV: Complex Integration ... Section 1: Riemann-Stieljes Integral ... ... I need help in fully understanding another aspect of the proof of Proposition 1.3...

I am reading John B. Conway's book, "Functions of a Complex Variable I" (Second Edition) ... I am currently focussed on Chapter IV: Complex Integration ... Section 1: Riemann-Stieljes Integral ... ... I need help in fully understanding aspects of Proposition 1.3 ...Proposition 1.3 and its...

Homework Statement Prove that {##x \epsilon \mathbb{R} : x^2 \ge 1##} is "not" bounded below. EDIT: I Looked closely and realized there is a "not" that we all had to write in...sorry if you lost some time.. Homework Equations Defintion: We say a nonempty subset ##A## of ##\mathbb{R}## is...

Let x(t) a positive function satisfied the following differential inequality $\frac{x'(t)}{1+{x(t)}^{2}}+x(t)f(t)<2f(t)$ , (1) with $0\leq t\leq T$ , $\arctan(0)<\frac{\pi }{2}$ and $f(t)$ is a positive function. Is x(t) bounded for all $T\geq 0$?

$\text{Evaluate } $ \begin{align*} I&=\iiint\limits_{E} x^2 e^y dV \end{align*} $\text{where E is bounded by the parabolic cylinder} $ \begin{align*} z&=1 - y^2 \end{align*} $\text{and the planes $z=0, x=1,$ and $x=-1$}\\$

Homework Statement Assume that ##a_k > 0## and ##\sum_{k=0}^\infty a_n## converges. Then for every ##\epsilon > 0##, there exists a ##n \in Bbb{N}## such that ##\sum_{k=n+1}^\infty a_k < \epsilon##. Homework EquationsThe Attempt at a Solution Since the series converges, the sequence of...

Homework Statement Given that ##\{x_n\}## is a bounded, divergent sequence of real numbers, which of the following must be true? (A) ##(x_n)## contains infinitely many convergent subsequences (B) ##(x_n)## contains convergent subsequences with different limits (C) The sequence whose...

ok so there are 3 peices to this Express and integral for finding the area of region bounded by: \begin{align*}\displaystyle y&=2\sqrt{x}\\ 3y&=x\\ y&=x-2 \end{align*}

1. The problem statement: Let ##f:[a, b] \rightarrow \mathbb{R}##. Prove that if ##f## is continuous, then ##f## is bounded. 2. Relevant Information This is the previous exercise. I have already proved this result, and the book states to use it to prove the next exercise. It also hints to use...

Homework Statement I have a question. I need to know the integral dxdydz/(y+z) where x>=0, y>=0, z>=0.Homework Equations It is bounded by x + y + z = 1. The transformations I need to use are x=u(1-v), y=uv(1-w), z=uvw. The Attempt at a Solution y+z = uv. J = uv(v-v^2+uv) So I get the integral...

Homework Statement I am trying to show that the integrator is unstable by giving examples of bounded inputs which produce unbounded outputs (i.e. a bounded function whose integral is unbounded). Note: The integrator is a system which gives an output equal to the anti-derivative of its input...

Homework Statement Find the maximum and minimum value attained by f(x, y) = x2 + y2 - 2x over a triangular region R with vertices at (0, 0), (2, 0), and (0, 2) Homework Equations partial x = 0 and partial y = 0 at extrema The Attempt at a Solution partial x = 2x - 2 partial y = 2y 2x - 2 =...

So the definition of a bounded sequence is this: A sequence ##(x_{n})## of real numbers is bounded if there exists a real number ##M>0## such that ##|x_{n}|\le M## for each ##n## My question is pretty simple. How does one choose the M, based on the sequence in order to arrive at the...

$\tiny{242t.08.02.41}$ $\textsf{Find the area of the region bounded above by}$ $\textsf{$y=8\cos{x}$ and below by $4\sec{x}$}$ $\textsf{and the limits are $-\frac{\pi}{4}\le x \le \frac{\pi}{4}$}$ \begin{align*} \displaystyle I_{41}&=\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} (8\cos{x}-4\sec{x})\,dx...

1. I have to show that S1 = {x ∈ R2 : x1 ≥ 0,x2 ≥ 0,x1 + x2 = 2} is a bounded set.2. So I have to show that sqrt(x1^2+x2^2)<M for all (x1,x2) in S1.3. I have said that M>0 and we have 0<=x1<=2 and 0<=x2<=2. And x2 = 2-x1 We can fill in sqrt(x1^2 + (2-x1)^2) = sqrt (0^2 + (2-0)^2) = 2 < M = 3...

Hey! :o Let $D\subseteq \mathbb{R}$ be a non-empty set. I want to show that $D$ ist compact if and only if each continuous function is bounded on $D$. I have done the following: We suppose that $D$ is compact. Since $f$ is continuous, we have that $f(D)$ is also compact, right? (Wondering)...

$\tiny{AP.6.1.1}\\$ $\textsf{Let $f(x)=x^3$}\\$ $\textsf{A region is bounded between the graphs of $y=-1$ and $y=\ f(x)$ }\\$ $\textsf{for x between $-1$ and $0$, region. }\\$ $\textsf{And between the graph of $y=1$ and $y=f(x)$ for x between $0$ and $1$ }\\$ $\textsf{This appears to be...

I am using Lang's book on complex analysis, i am trying to reprove theorem 4.1 which is a simple theorem: Let Compact(S \in \mathbb{C}) \iff Closed(S) \land Bounded(S) I will show my attempt on one direction of the proof only, before even trying the other direction. Assume S is compact Idea...

Calculate the area of the region bounded by the graph of the function y = 8 – 2x - x^2 and the x-axis Y = 8 - 2- x^2 0 = 8 – 2 – x^2 (-x – 4)(x – 2) - x – 4 = 0 and x – 2 = 0 -x = 4 x = 2 X = - 4 Do I do this? Y = 8 -2x -x^2 = 8x - (2x^2)/2 - x^3/3 = 8 -...

Homework Statement By using cylindrical coordinates , evaluate the volume of solid bounded on top of sphere (x^2) + (y^2) + (z^2) = 9 and it's sides by (x^2) + (y^2) = 4x . [/B]Homework EquationsThe Attempt at a Solution I have sketched out the diagram , but i dun know which part is the solid...

Homework Statement By using cylindrical coordinate , evaluate ∫ ∫ ∫ zDv , where G is the solid bounded by the cylinder (y^2) + (z^2) = 1 , cut by plane of y = x , x = 0 and z = 0 I can understand that the solid formed , was cut by x = 0 , thus the base of the solid formed has circle of (y^2) +...

Homework Statement Find the surface portion bounded by plane 2x +5y + z = 10 that lies in cylinder (x^2) +(y^2) = 9 ... I have skteched out the diagram and my ans is 5sqrt(30) instead of 9sqrt (30) as given by the author ... Anything wrong with my working ? Homework EquationsThe Attempt at a...

Homework Statement Find the surface area of portion of plane x + y + z = 3 that lies above the disc (x^2) + (y^2) < 2 in the first octant ... Homework EquationsThe Attempt at a Solution Here's the solution provided by the author ... I think it's wrong ... I think it should be the green...

Homework Statement On a sample midterm for my Calc 3 class the following question appears: Find the mass of (and sketch) the region E with density ##\rho = ky## bounded by the 'cylinder' ##y =\sin x## and the planes ##z=1-y, z=0, x=0## for ##0\le x\le\pi/2##. Homework Equations $$ m= \int_{E}...

Homework Statement [/B] Calculate the volume bounded by the plane/cylinder x^2+y^2=1 and the planes x+z=1 and y-z=-1. Homework Equations / The attempt at a solution[/B] It is pretty basic triple integral in cylindrical coordinates. For some reason, I can't get the right answer. I'm using...

It is well known that a curve in ##\mathbb{R}^3## is uniquely (up to a position in the space) defined by its curvature ##\kappa(s)## and torsion ##\tau(s)##, here ##s## is the arc-length parameter. We will consider ##\kappa(s),\tau(s)\in C[0,\infty)## Thus a natural problem arises: to restore...

Homework Statement FInd the area bounded by x=-3, y=-x^2-2x, and y=x^2-4. (Hint: Graph the picture) 2. The attempt at a solution My professor did set up the problem in class, but its throwing me off. He set it up as the lower bound -3 to 2, with the function (2x^2+2x-4)dx. I tried solving this...

Can there be a bounded space without a boundary without embedding in a higher spatial dimension? This seems to be the kind of question I get stuck on when the big bang comes up. Thanks

Is the subset ##E## necessary in the following definition? It doesn't seem to serve any purpose at all and could've been written with ##S## directly? Isn't ##E## just another ordered set since it's a subset of ##S##? Definition: Suppose ##S## is an ordered set, and ##E \subset S##. If there...

Homework Statement So I am trying to accomplish the above by using spherical coordinates, I am aware the problem may be solved using dv=dxdydz= zdxdy were z is known but I would like to try it using a different approach (using spherical coordinates). Any help would be greatly appreciated...

Homework Statement Find area bounded by parabola y^2=2px,p\in\mathbb R and normal to parabola that closes an angle \alpha=\frac{3\pi}{4} with the positive Ox axis. Homework Equations -Area -Integration -Analytic geometry The Attempt at a Solution For p>0 we can find the normal to parabola...

Homework Statement Find area bounded by functions y_1=\sqrt{4x-x^2} and y_2=x\sqrt{4x-x^2}. Homework Equations -Integration -Area The Attempt at a Solution From y_1=y_2\Rightarrow x=1. Intersection points of y_1 and [/itex]y_2[/itex] are A(0,0),B(1,\sqrt 3),C(4,0). Domain of y_1 and y_2 is...

A number of scientists subscribe to this theory. I read up on it, but none of the explanations I found really answered my questions. How should one attempt to envision a universe that is finite and bounded?

Hello! (Wave)We have that $S_{||\cdot||_2}:= \{ x \in \mathbb{R}^n: ||x||_2=1\}$. How can we justify that the above set is bounded? Do we just say that if $x \in S_{||\cdot||_2}$ then $||x||_2=1 \leq 1$ and so the set is bounded. How could we justify it more formally?

Homework Statement [/B] Use the Bounded Monotonic Sequence Theorem to prove that the sequence: \{a_{i} \} = \Big\{ i - \sqrt{i^{2}+1} \Big\} Is convergent.Homework EquationsThe Attempt at a Solution [/B] I've shown that it has an upper bound and is monotonic increasing, however it is to...

Why we sometimes take the area bounded by the curve is sum of positive area and absolute of negative area(e.g. ∫\int_0^2π sin(x)\, dx is equal to 4 or area of ellipse )?But sometimes we just sum positive and negative areas which is equal to 0(e.g. area of cycloid →when we integrate we get...

Assume that ##\{f_n\}## is a sequence of monotonically increasing functions on ##\mathbb{R}## with ## 0\leq f_n(x) \leq 1 \forall x, n##. Show that there is a subsequence ##n_k## and a function ##f(x) = \underset{k\to\infty}{\lim}f_{n_k}(x)## for every ##x\in \mathbb{R}##. (1) Show that some...

I figured it out... how do I remove this question?

Find the coordinate of center of mass. Given: The quarter disk in the first quadrant bounded by x^2+y^2=4 I tried to solve this problem but can't figure out how to do it. so y integration limit is: 0 <= y <= sqrt(4-x^2)) x limit of integration: 0 <= x <= 2 and then after the dy integral I...

Homework Statement Find the volume of the solid bounded by z=x^2+y^2 and z=8-x^2-y^2 Homework Equations use double integral dydx the textbook divided the volume into 4 parts, The Attempt at a Solution [/B] f(x)= 8-x^2-y^2-(x^2+y^2)= 4-x^2-y^2 i use wolfram and got 8 pi, the correct...

Find the volume of the solid in the first octant bounded by the parabolic cylinder z = 25 − x2 and the plane y = 2. I already solved it and got 710/3 as my answer, I just wanted to make sure its the right answer

Homework Statement Find the volume of the solid bounded by the surfaces ## (x^2 + y^2 + y)^2 = x^2 + y^2 ## ##x + y + z = 3 ## and ##z = 0## Homework EquationsThe Attempt at a Solution I begin by converting to polar coordinates to do a cylindrical integration with 3 variables. ## (x^2 + y^2 +...

Homework Statement Hi, I've been solving Calculus Deconstructed by Nitecki and I've been confused by a particular lemma in the book. Namely: If a sequence is eventually bounded, then it is bounded: that is, to show that a sequence is bounded, we need only find a number γ ∈ R such that the...

I don't have a particular problem in mind here, so please move this thread if it's in the wrong section. I was wondering, when you're trying to find the area bounded between two curves, is there a foolproof way to choose which curve to be g(x) in (let S be the integral sign, haha)...