Bounded Definition and 537 Threads

  1. C

    MHB Show that a sequence is bounded, monotone, using The Convergence Theorem

    Dear Every one, In my book, Basic Analysis by Jiri Lebel, the exercise states "show that the sequence $\left\{(n+1)/n\right\}$ is monotone, bounded, and use the monotone convergence theorem to find the limit" My Work: The Proof: Bound The sequence is bounded by 0. $\left|{(n+1)/n}\right|...
  2. Eclair_de_XII

    Prove: A bounded sequence contains a convergent subsequence.

    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...
  3. Jarvis323

    I Divisibility of bounded interval of reals

    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...
  4. S

    A Are bounded operators bounded indepedently on the function?

    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...
  5. S

    A Is this operator bounded or unbounded?

    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...
  6. M

    MHB Show that the function is bounded and strictly increasing

    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...
  7. Turbotanten

    A What does it mean for the Hamiltonian to not be bounded?

    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}...
  8. Math Amateur

    MHB Complex Valued Functions BV: John B. Conway Prop 1.3 Explained

    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...
  9. Math Amateur

    MHB Understand Proposition 1.3 in Conway's Functions of Complex Variables I

    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...
  10. F

    Can a set include negative infinity and be bounded below

    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...
  11. R

    MHB Bounded Solution For Differential Inequality

    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 $T\geq 0$?
  12. karush

    MHB Q2:2 Where E Is Bounded By The Parabolic Cylinder

    $\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 $x=-1$}\\$
  13. B

    Convergent Series Can Be Bounded by Any ##\epsilon>0##

    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...
  14. B

    Does a Bounded, Divergent Sequence Always Have Multiple Convergent Subsequences?

    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...
  15. karush

    MHB 15.3 Express an integral for finding the area of region bounded by:

    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*}
  16. Oats

    Why is a finite sub-cover necessary for proving continuity implies boundedness?

    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...
  17. C

    Integral with transformations and bounded by x + y + z = 1

    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...
  18. R

    Bounded functions with unbounded integrals

    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...
  19. Kaura

    Extrema of Two Variable Bounded Function

    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 =...
  20. M

    I For direct proof, how do you choose M for bounded sequence?

    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...
  21. karush

    MHB 242t.08.02.41 Find the area of the region bounded by

    $\tiny{242t.08.02.41}$ $\textsf{Find the area of the region bounded above by}$ $\textsf{$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...
  22. C

    How to prove a set is a bounded set?

    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...
  23. M

    MHB How could we show that D is bounded and closed?

    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)...
  24. karush

    MHB AP.6.1.1 A region is bounded between the graphs

    $\tiny{AP.6.1.1}\\$ $\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...
  25. Z

    I Proof check: S in C Compact implies S is closed and bounded

    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...
  26. N

    MHB What is the area bounded by y = 8 – 2x - x^2 and the x-axis?

    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 -...
  27. C

    Solid bounded by different region

    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...
  28. C

    Solid bounded by the cylinder (y^2) + (z^2) = 1 , cut by pla

    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) +...
  29. C

    Surface portion bounded by plane 2x +5y + z = 10 that lies

    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...
  30. C

    Surface area bounded by 2 different planes

    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...
  31. T

    Mass of Region Bounded by y=sin(x), z=1-y, z=0, and x=0

    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}...
  32. D

    Volume of cylinder bounded by two dependent planes, ideas?

    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...
  33. wrobel

    I What Is the Geometrical Interpretation of Bounded Curves?

    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...
  34. D

    How do I correctly find the area bounded by x=-3, y=-x^2-2x, and y=x^2-4?

    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...
  35. P

    I Can there be a bounded space w/o a boundary w/o embedding?

    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
  36. I

    I Why use a subset in the definition of bounded above?

    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...
  37. N

    Volume of a solid bounded by a paraboloid and the x-y plane

    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...
  38. N

    I Calculating total force over bounded area [Given p density]

    Hey, I am trying to prove that taking a 'horizontal' and 'vertical' strip equates to the same answer for the following problem. I have the current solution for taking a horizontal strip (ie dA = dxdy) and letting the bounds of x be between the two equations x(y) and the bounds of y be between...
  39. G

    Finding the Area Bounded by Two Functions

    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...
  40. G

    Find the area bounded by curves

    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...
  41. Alltimegreat1

    What is a finite and bounded universe and how do scientists envision it?

    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?
  42. evinda

    MHB Justifying Set Boundedness of $S_{||\cdot||_2}$ in $\mathbb{R}^n

    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?
  43. Euler2718

    Bounded Monotonic Sequence Theorem

    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...
  44. L

    Sum of area bounded by the curve

    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...
  45. R

    Subsequences of bounded monotonically increasing function

    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...
  46. T

    The quarter disk in the first quadrant bounded by x^2+y^2=4

    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...
  47. qq545282501

    Volume bounded by two surfaces, what am I missing?

    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...
  48. C

    MHB Volume of Parabolic Cylinder in 1st Octant: 710/3

    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
  49. M

    Volume bounded by 3 surfaces, did I do this correctly?

    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 +...
Back
Top