Bounded Definition and 537 Threads

  1. F

    Is f Bounded if It Has a Limit at Every Point on a Closed Interval?

    Homework Statement Let f:[a, b] -> R have a limit at each x in [a, b]. Prove that f is bounded. Homework Equations None The Attempt at a Solution No idea on how to start the proof. Completely lost. Thank you
  2. I

    Continuous, bounded, and not uniform?

    Homework Statement Give an example of a function f : R -> R where f is continuous and bounded but not uniformly continuous. Homework Equations A function f : D -> R and R contains D, with Xo in D, and | X - Xo | < delta (X in D), implies | f(X) - f(Xo) | < epsilon. Then f is continuous...
  3. G

    Prove that a sequence which is bounded above cannot tend to infinity

    question 1 : Prove that a sequence which is bounded above cannot tend to infinity What i did was state the definition ... but I'm trying to proof by contradiction. So i first suppose that a(n) tends to infinity , then a(n) > C . But since it is bounded above , C < or = to U , where U is the...
  4. S

    Does Every ε > 0 Assure a y in Y Such That x < y + ε?

    Homework Statement x is an upper bound for the set Y. Prove that x = sup(Y) (least upper bound) if and only if for every e > 0, there is some y in Y (depending on e) such that x < y + e (e in this case is every positive real number) The Attempt at a Solution since for every y in...
  5. J

    Proving Boundedness of Set S: |x| + |y| <= 2

    Is the set S = {(x,y): |x| + |y| <= 2} bounded? If so how do i prove it? looking at the graph i believe that S is bounded by 2 and -2, but I'm not sure if I'm correct and i don't know how to prove it. thanks!
  6. soarce

    Bounded solutions of a system of copled liniar Schrodinger equations

    bounded solutions of a system of coupled liniar Schrodinger equations Hi. I study the following system of four coupled liniar Schrodinger equations: i\delta \left(\begin{array}{c}f&h&g&q \end{array}\right) = \left(\begin{array}{cccc} -L_p&-a_1&-a_2&-a_2\\ a_1&L_p&a_2&a_2\\...
  7. D

    Find the area bounded by the parabolas

    Homework Statement find the area bounded by the parabolas y=2x^2-x-15 and y=x^2-4x-5 Homework Equations The Attempt at a Solution x^2+3x-10=0 I got x = 2 and x = -5; is that right? If so, why do I keep getting an area of 76.17 when I integrate from -5 to 2? I end up with...
  8. K

    Prove that a converging sequence is bounded

    Homework Statement Suppose that the sequence {an}converges. Show that the sequence {an} is bounded. The Attempt at a Solution Since the sequence converges, for every delta>0, there must exist a number N such that for every n>=N, |an - x|< delta. Therefore, for n>=N, -delta+x < an < delta...
  9. A

    Condition for a 2nd order differential eqn to have bounded solutions?

    Suppose I have a 2nd order differential equation a_1y''(x)+a_2y'(x)+a_3y(x)+a_4=0 and two conditions y(0), y'(0). Then is there any theorem which gives us the condition under which the solution y(x) will be bounded? Note that x-range is entire real line. This is a general version of the...
  10. E

    Traveling from Point A to Point B at the Speed of Light

    If I were to travel from point a to point b at the speed of light, given that a and b are real, I would travel a constant distance through space, and because space is bonded with time, traveling from point a to point b at the speed of light would result in a constant distance in time. It may...
  11. T

    Definite intergration area under curve bounded with line

    Homework Statement A cruve has the equation y = x{3} - 8x^{2} + 20x . The curve has stationary points A and B. There is a line through B parallel to y-axis and meets the x-axis at the point N. The region R is bounded by the curve , the x-axis and the line from A to N. Find the exact area under...
  12. J

    Making given metric space bounded

    I just encountered a claim, that for any given metric space (X,d), there exists another topologically equivalent metric d' so that (X,d') is bounded. Anyone knowing anything about the proof for this?
  13. B

    Is the Inverse Image of a Compact Set Always Bounded?

    Homework Statement Let f be a continuous mapping from metric spaces X to Y. K \subset Yis compact. Is f^{-1}(K) bounded? Homework Equations Theorem 4.8 Corollary (Rudin) A mapping f of a metric space X into Y is continuous iff f^{-1}(C) is closed in X for every closed set C in Y...
  14. R

    Bounded Probability Density Function

    Let the random variable X have the probability density function f(x). Suppose f(x) is continuous over its domain and Var[X] is bounded away from zero: 0 < c < Var[X]. Claim: f(x) is bounded over its domain. Is this claim true? I don't think a counterexample like X ~ ChiSq_1 applies...
  15. R

    Exploring the Bounded Universe: Evidence and Controversies

    If the universe is bounded then there will be an absolute frame of reference. This could be found by summing all the frames of reference up to the unltimate one ie your in a car traveling on road that's on the earh which is spinning on it axis and simutaneously moving about the sun . The sun...
  16. B

    Evaluating Integral for Region Bounded by x^2 - xy + y^2 = 2

    1. Find the area of the region bounded by x^2 - xy + y^2 = 2: a)let x = au + bv, y= au - bv therefore, 3b^2v^2 + a^2u^2 = 2 b) Choose a and b such that u^2 + v^2 = 1, therefore, a = sqrt 2 & b = (sqrt 6)/3 c) Applying these results and changing variables into u and v, evaluate the...
  17. P

    Are compact sets in an arbitrary metric space always bounded?

    Homework Statement Prove that every compact set is bounded. Homework Equations The usual compactness stuff - a compact set in a metric space X is one that, for every open cover, there is a finite subcover. The Attempt at a Solution I'm really hesitant about this question because my...
  18. S

    Prove: Bounded Set A U B is Bounded

    Homework Statement If A and B are bounded sets, then A U B is a bounded set. (Prove this statement) Homework Equations Definition of Union is a given. A set A is bounded iff there exists some real value m such that lxl < m for all element x found in A. The Attempt at a...
  19. D

    Average of Log of a Function: Bounded by 1 and Convex

    Hello, I am interested in the average behaviour of the log of a function. I know the average of the function over the range of interest: F = \frac{1}{(b-a)} \int_a^b f(x) dx. I also know that f(x) is convex and bounded from below by 1. I want to know the average \frac{1}{(b-a)}...
  20. B

    Hilber space and linear bounded operator

    Let H be a Hilbert space and A: H-> H be a Linear Bounded Operator. Show that A can be written as A=B+C where B and C are Linear Bounded Operators and B is self-adjoint and C is skew. This is suppose to be an easy question but I'm not sure where to start. I know that self-adjoint is (B*=B)...
  21. E

    If f is continuous on [a,b], then f is bounded on [a,b].

    Dear friends, I just joined the forums, and I'm looking forward to being a part of this online community. This semester, I signed up for Analysis II. I'm a math major, so I should be able to understand pretty much everything you say (hopefully). However, I'd really appreciate it if you try...
  22. E

    If f is continuous on [a, b], then f is bounded on [a,b].

    Dear friends, I just joined the forums, and I'm looking forward to being a part of this online community. This semester, I signed up for Analysis II. I'm a math major, so I should be able to understand pretty much everything you say (hopefully). However, I'd really appreciate it if you try...
  23. S

    When is a function bounded using differentiation

    Homework Statement how do i determine whether a function is bounded using differentiation eg: f(x)=x/(2^x) Homework Equations The Attempt at a Solution i know it has something to do with maximums and minimums but i can't figure out how to do it. any help would be appreciated...
  24. S

    Solving Bounded Sequences Homework - How to Find Bounding Number

    Homework Statement how do show whether the following sequences are bounded? 1) {an}=sqrt(n)/1000 2) {an}=(-2n^2)/(4n^2 -1) 3) {an}=n/(2^n) 4) {an}=(ncos(npi))/2^n Homework Equations i have to show whether the sequences are bounded by a number but i don't know how to find that number...
  25. S

    Determine if a sequence {an} is monotonic, bounded, convergent

    Determine whether the sequence {an} defined below is (a) monotonic (b) bounded (c) convergent and if so determine the limit. (1) {an}=(sqrt(n))/1000 a) it is monotonic as the sequence increase as n increases. b) it's not bounded (but I'm not sure why) c) divergent since limit doesn't...
  26. R

    Set of continuous bounded functions.

    Homework Statement This is not a homework question. I am solving this from the lecture notes that one of my friend's has got from last year. If C(X) denotes a set of continuous bounded functions with domain X, then if X= [0,1] and fn(x) = x^n. Does the sequence of functions {fn} closed ...
  27. R

    Every sequence of bounded functions that is uniformly converent is uniformly bounded

    Homework Statement Prove that every sequence of bounded functions that is uniformly convergent is uniformly bounded. Homework Equations Let {fn} be the sequence of functions and it converges to f. Then for all n >= N, and all x, we have |fn -f| <= e (for all e >0). ---------- (1)...
  28. X

    Maximal area of a convex region bounded by hyperbolas

    This problem was suggested by Gokul43201, based on this year's Putnam A2. Suppose that K is a convex set in \mathbb{R}^2 which is contained in the region bounded by the graphs of the hyperbolas xy=1, xy=-1 (so the set is in the inner + shaped region which contains the origin also). What is...
  29. B

    Area Bounded by a Parabola and Two Lines: Calculating with Riemann Sums?

    Homework Statement Find the area bounded by y = 3x^2 + 1, x = 0, x = 2, y = 0 Homework Equations The Attempt at a Solution Not sure how to do this? Is this like finding the upper and lower sums?
  30. V

    Inverse Image of a Compact Set - Bounded?

    [SOLVED] Inverse Image of a Compact Set -- Bounded? Problem: Let f : X → Y be a continuous function, K ⊂ Y - compact set. Is it true that f^{-1}(K)– the inverse image of a compact set– is bounded? Prove or provide counterexample. Questions Generated: 1. Why does compactness matter? (I...
  31. R

    Variational calculus with bounded derivative constraints

    [SOLVED] Variational calculus with bounded derivative constraints After learning about the calculus of variations and optimal control for a bit this semester, I've decided to tackle a "simple" (in the words of my professor) problem meant to illustrate a simplified example of highway...
  32. Z

    Is a Monotone Sequence with a Bounded Subsequence Always Bounded?

    Homework Statement prove that a monotone sequence which has a bounded subsequence is bounded Homework Equations The Attempt at a Solution
  33. C

    Are Sequence Definitions for Bounded Sequences Adequate for Proving Boundedness?

    Definitions: Let {x[n]} be a bounded sequence in Reals. We define {y[k]} and {z[k]} by y[k]=sup{x[n]: n \geq k}, z[k]=inf{x[n]: n \geq k} Claim: (i) Both y[k] and z[k] are bounded sequences (ii){y[k]} is a decreasing sequence (iii){z[k]} is an increasing sequence Proof: (i)...
  34. B

    Double integral of volume bounded by plane and paraboloid

    Evaluate the volume of the solid bounded by the plane z=x and the paraboloid z = x^2 + y^2 I have tried to graph this, and they don't bound anything? have i graphed it wrong. and is there a way to do these problems where you don't need to draw the graph.
  35. S

    Bounded sequence implies convergent subsequence

    How can you deduce that nad bounded sequence in R has a convergent subsequence?
  36. V

    Convergence and Cauchy Sequences in Rational Numbers

    Homework Statement Prove that if {a_{n}} is a sequence of rational numbers such that {a_{n+1}} > {a_{n}} for all n \in \textbf{N} and there exists an M\in \textbf{Q} such that {a_{n}} \leq M for all n \in \textbf{N}, then {a_{n}} is a Cauchy sequence of rational numbers.Homework Equations Do...
  37. B

    Iterated Integrals bounded by curves

    Evaluate \int\int_{Q}\left(1 - x^{3}\right)y^{2} dA where Q is the region bounded by y=x^2 and x = y^2 So I have drew the graphs of y=x^2 and x=y^2 and found that they intersect at (0,0) and (1,1). Now I am confused what to replace Q with, but I think it should be this: please tell me if I am...
  38. tony873004

    Find the areas bounded by 4 equations

    Homework Statement I won't post the entire problem since I'm only stuck on one part of it. I need to find where y=cos x and y=sin 2x intersect. Homework Equations sin(x)=cos(x +- pi/2) The Attempt at a Solution cos x = sin 2x since sin(x)=cos(x +- pi/2), sin 2x =...
  39. K

    Difference between closed set and bounded set

    The way they use the terms:"closed set" and "bounded set" make me thinking that a closed set is different from a bounded set but i can not figure out how to prove that. So can some body show me clearly the difference between those two terms?
  40. P

    Monotonic bounded sequence theorem

    So the theorem states if a sequence is monotonic and bounded, it converges. WEll, it's easy enough to prove is a sequence is monotonic, but how would one go about proving that a sequence is bounded?
  41. daniel_i_l

    Uniform continuity with bounded functions

    Homework Statement True or false: 1)If f is bounded in R and is uniformly continues in every finate segment of R then it's uniformly continues for all R. 2)If f is continues and bounded in R then it's uniformly continues in R. Homework Equations The Attempt at a Solution 1) If...
  42. W

    Bounding Region Inequalities for Solid Rectangular Box in First Octant

    Homework Statement Ok I just wanted to make sure of this one. Write inequalities to describe the region: The solid rectangular box in the first octant bounded by the pane x=1 y=2 z=3. The Attempt at a Solution I thought of it as the volume of the box bounded boy the planes so: 0\leq...
  43. R

    Uniform continuity, bounded subsets

    Homework Statement Show that if f: S -> Rn is uniformly continuous and S is bounded, then f(S) is bounded. Homework Equations Uniformly continuous on S: for every e>0 there exists d>0 s.t. for every x,y in S, |x-y| < d implies |f(x) - f(y)| < e bounded: a set S in Rn is bounded if it is...
  44. P

    Totally bounded & Heine Borel?

    Homework Statement Why doesn't a set being totally bounded imply the set has the Heine Borel property? Another related question is what happens if a cover consist of open balls that cover the set and more of it? i.e. A=(-1,2) U (1,3) covers (0,1) but really covers more than (0,1). Is A...
  45. P

    Cauchy Boundedness: Partial Sums Unbounded?

    Homework Statement Thm: If a sequence is Cauchy than that sequence is bounded. However Take the partial sums of the series (sigma,n->infinity)(1/n). The partial sums form a series which is Cauchy. But the series diverges so the sequence of partial sums is unbounded. Sequence of partial sums...
  46. R

    A question on bounded linear operators (Functional Analysis)

    Suppose T: X -> Y and S: Y -> Z , X,Y,Z normed spaces , are bounded linear operators. Is there an example where T and S are not the zero operators but SoT (composition) is the zero operator?
  47. S

    Show Set is Bounded: Prove A is Bounded in Q

    Hi everyone. I'm a math student still learning to do proofs. Here is a problem I encountered that seems easy but has me stuck. 1. The problem statement, all variables and given/known data Let a be a positive rational number. Let A = {x e Q (that is, e is an element of the rationals) |...
  48. C

    Finding Areas of Regions Bounded by Trig Functions Using Integrals

    Homework Statement Find the exact total of the areas bounded by the following functions: f(x) = sinx g(x) = cosx x = 0 x = 2pi Homework Equations the integral of (top equation - bottom equation) The Attempt at a Solution Change the window on the graphing calculator to...
  49. tony873004

    Rotate Area Bounded by y=5, y=x+(4/x) about x=-1: Verify Limits of Int.

    Rotate the area bounded by y = 5,\,y = x + (4/x) about x=-1 Verify the limits of integration x + \left( {4/x} \right) = 5,\,\,x = 1\& 4 solve \begin{array}{l} \int\limits_1^4 {2\pi r\,h\,dx} \\ h = 5 - \left( {x + \left( {4/x} \right)} \right),\,\,r = x + 1 \\ 2\pi...
  50. K

    Positive integral implies bounded below?

    Homework Statement The integral of f on [a,b] exists and is positive. Prove there is a subinterval J of [a,b] and a constant c such that f(x) >= c > 0 for all x in J. Hint: Consider the lower integral of f on [a,b] Homework Equations The Attempt at a Solution I don't see how...
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