Bounded Definition and 514 Threads

Homework Statement fn is a sequence of functions and sn is a sequence of reals such that 0 ≤ fn(x) ≤ sn for all x. I want to show that if \sum_{k=0}^{n}s_k is Cauchy then \sum_{k=0}^{n}f_k is uniformly Cauchy and that if \sum_{k=0}^{\infty}s_k converges then \sum_{k=0}^{\infty}f_k converges...

Homework Statement f(x) = (x^3) + (x^2) - (x) g(x) = 20*sin(x^2) Homework Equations The Attempt at a Solution I found the zeroes of the two functions at 4 intersections, and then the zeroes of each function respectively (there's 3 for f(x) and 4 for g(x) between -3 and 3), for certain...

Homework Statement Find the region bounded by the two functions from y=0 to y=2 equations given: x=(y-1)2 -1 x=(y-1)2 +1 express x as a function of y and integrate it with respect to y Homework Equations equations given: x=(y-1)2 -1 x=(y-1)2 +1 The Attempt at a Solution...

So this is more so a general question and not a specific problem. What exactly is the diefference between closed and boundedness? So the definition of closed is a set that contains its interior and boundary points, and the definition of bounded is if all the numbers say in a sequence are...

Hello, I need help with 2 homework questions: Also this question:

Homework Statement Suppose that f is a bounded, increasing function on [a,b]. If p is the partition of [a,b] into n equal sub intervals, compute Sp - sp and hence show f is integrable on [a,b]. What can you say about a decreasing function?Homework Equations We partition [a,b] into...

1. Ok, so the question is.. Find the exact volume of the solid bounded above by the surface z=e^{-x^2-y^2}, below by the xy-plane, and on the side by x^2+y^2=1. 2. Alright. So, I know that I can use a double integral to find the volume, and switching to polar coordinates would be simpler...

Hi guys, I would like to understand why a circle (and in general a n-sphere) as a subset of R^2 (in general R^(n+1)) with the standard topolgy is considered a closed and a bounded set. I think that this can be a closed set because its complement (the interior of the circle and the rest of...

Hello everyone! I'm asked to find a set that is bounded and that has exactly two limit points, now this is how I am thinking. Consider the set $A_n = [0,\frac{1}{n}) \cup(2-\frac{1}{n},2]$, if $A_1 = [0,1)\cup(1,2]$, $A_2=[0,1/2)\cup (3/2,2]$. If I let $n$ grow indefinitely, I will have only...

Homework Statement Define a function f:ℝ->ℝ that is increasing, bounded, and discontinuous at every integer. Homework Equations The Attempt at a Solution I've tried defining a fuction using the greatest integer function but I cannot get it to be bounded with jump discontinuities...

In the diagram, the shaded region is bounded by the parabola y = x2 + 1, the y-axis and the line y = 5. Find the volume of the solid formed when the shaded region is rotated about the y-axis. Got no diagram but limits will be 2-0 coz its on right side

Homework Statement Find absolute maxima and minima of the function in the given region: T(x,y) = x2 + xy + y2 - 6x Region: Rectangular plate given by: 0 ≤ x ≤ 5, -3 ≤ y ≤ 3 Homework Equations First derivative test, fx =0, fy = 0 Second derivative test, fxxfyy - fxy2 = ? The...

Find the global max/min for z=xy^2 - 5 on the region bounded by y=x and y=1-x^2 in the xy-plane.

Homework Statement I need to prove that a closed ball(radius r about x0) is closed and bounded. The same goes for a sphere(radius r about x0). Homework Equations The Attempt at a Solution How does one go about proving something is closed and bounded? My book is not very helpful...

(a) Show that the function g(x) =[3 + sin(1/x-2)]/[1 + x^2] is bounded. This means to find real numbers m; M is an lR such that m ≤ g(x) ≤ M for all x is an lR (and to show that these inequalities are satisfied!). (b) Explain why the function: f(x) = { [x-2] [3 + sin(1/x-2)]/[1 + x^2] ...

Spivak's proof of "A closed bounded subset of R^n is compact" Hi guys, I'm currently taking a differential geometry course and decided I would read Spivak's Calculus on Manifolds, and then move on to his Differential Geometry series. There's a proof in here that feels unjustified to me, so...

I was attempting to find a counterexample to the problem below. I think I may have, but was ultimately left with more questions than answers. Consider the space, L, of all bounded sequences with the metric \rho_1 \displaystyle \rho_1(x,y)=\sum\limits_{t=1}^{\infty}2^{-t}|x_t-y_t| Show that a...

Homework Statement Let M be a metric space and A\subseteqM be any subset: Prove that if A is contained in some closed ball, then A is bounded. Homework Equations Def of closed-ball: \bar{B}R(x) = {y\inM:d(x,y)≤R} for some R>0 Def of bounded: A is bounded if \existsR>0 s.t. d(x,y)≤R...

Prove that the following are equivalent: a) A is bounded, b) A is "in" a closed ball Homework Statement The full problem is: Let M be a metric space an A\subseteqM be any subset. Prove that the following are equivalent: a)A is bounded. b)A is contained in some closed ball c)A is contained in...

Thomas-Finney defines a bounded sequence as follows: - A sequence an is said to be bounded if there exists a real number M such that |an| ≤ M for all n belonging to natural numbers. This is equivalent to saying -M ≤ an ≤ M So, if all terms of a sequence lies between, say -1 and 1, i.e...

Homework Statement Find the volume of the region D in R^3 which is inside the sphere x^2 + y^2 + z^2 = 4 and also inside the cone z = sqrt (x^2 + y^2) Homework Equations The Attempt at a Solution So I decided that the best approach might be finding the area under the sphere and...

Homework Statement f(x)=x2-1 and f(x)=2x+2 Homework Equations The Attempt at a Solution Points of intersection are -1 and 3. So you integrate using those as upper and lower and plug it in and subtract, right? But I get 0 for each. So nothing to subtract and 0 is not the correct...

Homework Statement Set up triple integrals for the integral of f(x,y,z)=6+4y over the region in the first octant that is bounded by the cone z=(x^2+y^2), the cylinder x^2+y^2=1 and the coordinate planes in rectangular, cylindrical, and spherical coordinates. Homework Equations...

Homework Statement Find the volume of the region bounded by the cylinder x^2 + y^2 =4 and the planes z=0, and x+z=3. Homework Equations V = ∫∫∫dzdxdy V=∫∫∫rdrdθ The Attempt at a Solution Alright, so I feel as though I'm missing a step somewhere along the way, but here's what I've gotten...

Q. If each individual function is bounded and if \(f_n\longrightarrow f \) uniformly on S, then prove that {fn} is uniformly bounded on S. Proof : Since each fn is bounded implies \(f_n \leq M_n\) \(\Longrightarrow f_1\leq M_1, f_2 \leq M_2​,\) and so on If M = max {M1, M2,...Mn } then each term...

Suppose I have a C∞ function, which I wish to prove attains its maximum/minimum. First I must prove that the function is bounded at all. If I determine R, the region (of the plane in this case) where the function is strictly positive, and integrate over R to find a finite answer, can I say the...

Homework Statement Find the mass of a solid of constant density that is bounded by the parabolic cylinder x=y2 and the planes x=z, z=0, and x=1. The Attempt at a Solution https://dl.dropbox.com/u/64325990/Photobook/Photo%202012-06-07%202%2033%2024%20PM.jpg I first drew some diagrams to...

Hi everybody, I am trying to solve the following problem and I get stuck on the last question. I would appreciate a lot that someone helps me . Here is the problem: Let D be the region bounded from below by the cone z= the root of (x^2 + z^2), and from above by the paraboloid z = 2 – x^2 –...

Homework Statement Let H be an \infty-dimensional Hilbert space and T:H\to{H} be an operator. Show that if T is compact, bounded and has closed range, then T has finite rank. Do not use the open-mapping theorem. Let B(H) denote the space of all bounded operators mapping H\to{H}, K(H) denote...

Homework Statement Let G = { f \in C[0,1] : ^{0}_{1}\int|f(x)|dx \leq 1 } Endowed with the metric d(f,h) = ^{0}_{1}\int|f(x)-h(x)|dx. Is G totally bounded? Prove or provide counterexample 2. Relevant Theorems Arzela-Ascoli Theorem, Theorems relating to compactness, equicontinuity etc...

Homework Statement ...Bounded by graphs of equations: z=xy, z=0, y=x, x=1 I don't know what z=xy is. The rest of boundaries are clear. I assume that when y=1 and x=1, z=1. But, is this a z=1 plane? Check my figure attached. Thank you. Homework Equations The Attempt at a Solution

Compact --> bounded In lecture 8 of Francis Su's Real Analysis online lecture series, he has a proof that a compact subset of a metric space is bounded: Given a metric space (X,d), if A is a compact subset of X, then every open cover of A has a finite subcover. Let B be a set of open balls of...

Homework Statement The operator T maps from L^p(-2,2)\rightarrow L^p(-2,2) is defined (Tf)(x) = f(x) x Show that the operator maps from L^p(-2,2) into the same. Homework Equations p is a natural from 1 to infinity. Holders inequality Substitution integrals The Attempt at a Solution I look at...

1. Evaluate the integral ∫VxdV inside domain V, where V is bounded by the planes x=0, y=x, z=0, and the surface x2+y2+z2=1 Answer given: 1/8 - √2/16 (which is NOT what I got.. ) 2. The attempt at a solution Ok, it's a triple integral, I know this. ∫dx runs from 0 to 1 ∫dy...

Homework Statement Find the volume of the solid bounded by the parabolic cylinder y = x^2 and the planes z = 3-y and z = 0Homework Equations The Attempt at a Solution Obviously, a triple integral must be used in the situation. Our professor never explained how to find the limits of...

Claim (?): limsup xk < ∞ k->∞ IF AND ONLY IF the sequence {xk} is bounded above. Does anyone know if this is true or not? (note that the claim is "if and only if") If it is true, why? Thanks!

Homework Statement Find the volume of the region bounded by the planes 7x + 6y + 8z = 9, y = x, x = 0, z = 0. Homework Equations Multiple integration. The Attempt at a Solution My attempt at a solution is attached. To test, I computed the answer with Wolfram Alpha which yielded an...

Homework Statement Show that if f: [a,b]→Re is increasing and the range of f is a bounded interval then f is continuous. Homework Equations N/A The Attempt at a Solution I have no idea where to start, but I decided to start with a couple of things. Proof: Let f: [a,b]→Re...

Homework Statement Prove that if F is uniformaly continuous on a bounded subset of ℝ, then F is bounded on A. Homework Equations The Attempt at a Solution F is uniformaly continuous on a bounded subset on A in ℝ. Therefore each ε>0, there exists δ(ε)>0 st. if x, u is in A where...

Homework Statement Suppose f is differentiable in \mathbb{C} and |f(z)| \leq C|z|^m for some m \geq 1, C > 0 and all z \in \mathbb{C} , show that; f(z) = a_1z + a_2 z^2 + a_3 z^3 + ... a_m z^m Homework EquationsThe Attempt at a Solution I can't seem to show this. It does the proof...

Homework Statement I would like to show that a weakly convergent sequence is necessarily bounded. The Attempt at a Solution I would like to conclude that if I consider a sequence {Jx_k} in X''. Then for each x' in X' we have that \sup|Jx_k(x')| over all k is finite. I am not sure why...

1. The region bounded by the given curves is rotated about the specific axis. Find the volume of the resulting solid by any method (disc or shell). 2. x^2 + (y-1)^2 = 1 3. This is my first time posting up a homework question, so I apologize if I didn't get the notation down...

Homework Statement to prove that a function bounded on [a,b] with finite discontinuities is Riemann integrable on [a,b] Homework Equations if f is R-integrable on [a,b], then \forall \epsilon > 0 \exists a partition P of [a,b] such that U(P,f)-L(P,f)<\epsilon The Attempt at a...

Homework Statement I have equations that are y1= 2sin(\frac{3}{2}x) and y2= \frac{1}{3}x the point where they intersect is called "a" (about x≈1.88). Find the center of mass where M is the total mass of the object.Homework Equations xcm= \frac{1}{M}∫x dM The Attempt at a Solution I found...

Homework Statement Consider the vector space l_infty(R) of all bounded sequences. Decide whether or not the following norms are defined on l_infty(R) . If they are, verify by axioms. If not, provide counter example. Homework Equations x in l_infty(R); x=(x_n), (i) || ||_# defined by...

Is there such thing as a bounded topological space? Or does 'boundedness' only apply to metric spaces?

Homework Statement Show that D = { (x,y,z) \in \mathbb{R}^{3} | 7x^2+2y^2 \leq 6, x^3+y \leq z \leq x^2y+5y^3} is bounded. Homework Equations Definition of bounded:D \subseteq \mathbb{R}^{n} is called bounded if there exists a M > 0 such that D \subseteq \{x \in \mathbb{R}^{n} | ||x|| \leq...

Find an $\displaystyle f(x)$ such that $\displaystyle \frac{1}{f(x)}$ is defined for all $\displaystyle x$ and is bounded, but $\displaystyle f(x)$ is decreasing.

I'm trying to show that continuous f : [a, b] -> R implies f uniformly continuous. f continuous if for all e > 0, x in [a, b], there exists d > 0 such that for all y in [a, b], ¦x - y¦ < d implies ¦f(x) - f(y)¦ < e. f uniformly continuous if for all e > 0, there exists d > 0 such that for...

Homework Statement Let f : R -> R be a continuous function such that f(0) = 0. If S := {f(x) | x in R} is not bounded above, prove that [0, infinity) ⊆ S (that is, S contains all non-negative real numbers). Then find an appropriate value for a in the Intermediate Value theorem...