Bounded Definition and 537 Threads

Homework Statement Prove that the following function is continuous and bounded in R+ Homework Equations f(x) = 1 + \int_0^x e^{-t^2} f(xt) dt \qquad \forall x \geq 0 The Attempt at a Solution I thought of using Taylor Formula, but the integral is t instead of x, so now i have...

Hey all, We were discussing bounded and unbounded sets in class, and looking over my notes, I see that I have some trouble understanding the concept. Here are three examples that our professor gave us: Set A = {x\inR | |x| <10} Set A = {x\inR | x<10} A\subseteqZ s.t. x~y iff x|y Set A =...

Homework Statement Prove that a bounded subset of R is totally bounded. Homework Equations The Attempt at a Solution Fix E > 0. Let A be subset of R, x be contained in A, and B(E/2, a) where E/2 is the radius of the ball and a is the center. Assume that B(E/2, a) is closed...

"if a vector field has only nondegenerate zeros then the number of zeros is bounded" With no idea how to show that without using Poincare-Hopf Theorem. Any proof possible without using any concept from algebraic topology?

Homework Statement Let property 1 be : If [ai,bi] is a sequence of intervals that are closed such that for each i the interval [a(i+1), b(i+1)] is either the left half of [ai,bi] or the right half, then there exists precisely 1 number in all intervals sequence. Show if a field f...

Homework Statement Prove that SU(n) is closed and bounded Homework Equations The Attempt at a Solution So in order to prove this, I first mapped SU(n) to be a subset of R^{{2n}^2}. To prove the closed portion, I tried mapping a sequence in SU(n) to a sequence in R^{{2n}^2}. However, I...

Here's the problem: Let f:D-R and c in R be and a accumulation point of D, which is a subset of R. Suppose that a<=f(x)<=b for all x in D, x not equal to c, and suppose that limx\rightarrowc f(x) = L. Prove that a<=L<=b. I'm having trouble here. I've tried to prove by contradiction, by...

Homework Statement Let (sn) be a sequence in R that is bounded but diverges. Show that (sn) has (at least) two convergent subsequences, the limits of which are different. Homework Equations The Attempt at a Solution I know that a convergent subsequence exists by...

Homework Statement Asssume (an) is a bounded sequence with the property that every convergent subsequence of (an) converges to the same limit a. Show that (an) must converge to a. Homework Equations The Attempt at a Solution If the subsequence converges to a we have , we have...

Homework Statement Let A = {x: x^2 + 3x + 2 <0}. Prove that this set is non-empty and bounded above. What is the least upper bound? Is it bounded below? The Attempt at a Solution Well, solving for the zeros and understanding that between the zeros, we satisfy our values of x's, I have...

Homework Statement let |e-x-e-y| be a metric, x,y over R. let X=[0,infinity) be a metric space. prove that X is closed, bounded but not compact. Homework Equations The Attempt at a Solution there is no problem for me to show that X is closed and bounded. but how do I prove...

Homework Statement Show that a sequence {a_n} is bounded if and only if there is an interval [c, d] such that {a_n} is a sequence in [c,d]. Homework Equations A sequence {a_n} is bounded provided that there is a number M such that |a_n| <= M. The Attempt at a Solution...

Question about the "bounded" metric Homework Statement I'm trying to show that the topology induced on \mathbb R by the bounded metric \sigma(x,y) = \left| \frac{x}{1+|x|} - \frac{y}{1+|y|} \right| is equivalent to that induced by the standard Euclidean metric d(x,y) = |x-y|. Homework...

Take the discreet metric on an infinite set A. I understand that its closed (because it contains all of its limit points), but I don't understand why its bounded and why its not compact. Also, when they say "an infinite set A" do they mean a set that extends to infinite (say, [1,n] for...

Homework Statement This is from baby rudin: If E\subset X and if f is a function defined on X, the restriction of f to E is the function g whose domain of definition is E, such that g(p)=f(p) for p\in E. Define f and g on R^2 by: f(0,0)=g(0,0)=0, f(x,y)=\frac{xy^2}{x^2+y^4}...

Its a question that I had from a friend in the past. I had tried solving it but to no avail. Have tried integration and stuff like that, but I think there is an easier way to solve this question. Question -> Square of 7cm, find the shaded area...

Hello all, I am a bit confused by the concept of "bounded almost surely". If a random variable X(\omega) is bounded a.s., so this means (i) X \leq K for some constant K ? or some K(\omega) ? Also, if it is bounded almost surely, does that mean it is also bounded in L^{p} ? Apparently if...

I have a sequence where s_1 can take any value and then s_{n+1}=\frac{s_n+10}{s_n+1} How do I show a sequence like this is bounded?

Having a hard time understanding this example from a book: The function f(x) = 1/x is locally bounded at each point x in the set E = (0,1). Let x \in (0,1). Take \delta_x = x/2, M_x = 2/x. Then f(t) = 1/t <= 2/x = M_x if x/2 = x-\delta_x < t < x + \delta_x This argument is false since...

Homework Statement Find the average value of z for a the spherical surface of radius R that resides above the x-y plane. Homework Equations Equation of a sphere x^2+y^2+z^2 = R^2 The Attempt at a Solution I rearrange the equation above and do a double integral z_{total} = \int...

Homework Statement 1. f(z) is a function that is analytic on all of the complex plane, and mod(f)<=mod(z). Prove that f=cz. 2. f(z) is analytic on all of the complex plane, and mod(f)<= sqrt(mod(z)). Prove that f is constant Homework Equations Liouvilles thm: the only bounded entire...

Homework Statement A = \left\{(x,y): 0\leq xy \leq 1\right\}, A \in R^{2} I'm trying to determine if this set is bounded and/or closed. Homework Equations if X = (x,y) euclidean metric: ||X|| = \sqrt{x^{2}+y^{2}} The Attempt at a Solution I know a bounded set =>...

Let f is monotone increasing, bounded, and differentiable on (a,inf) Then does it necessarily follow that lim(f'(x),x,inf)=0 ? It is hard to guess intuitively or imagine a counterexample...

Homework Statement find the Area Bounded by the two curves, y=|x+1|, y= - ( x+1)2 + 6 Homework Equations y=|x+1|, y= - ( x+1)2 + 6 The Attempt at a SolutionA= Integration of | f (x) - g(x) | x+1= f(x) -(x+1)2 + 6= g(x) getting the limit of integration: x+1= - (x+1)2 + 6 x2 + 3x - 4=0...

Homework Statement Let f ba analytic function on 0< |z| < 1 and suppose |f(z)| <= 4|z|^1.1 for all 0<|z|<1. Prove that |f(1/2)| <= 1 Homework Equations The Attempt at a Solution I tried to prove it be cauchy integral formula but I got |f(1/2)|< 8 r ^1.1 r<1

Homework Statement I have a problem where the graph of the equation is below the x-axis at x=0, crosses the x-axis at x=2, and is above at x=5. To find the area of this would I just split the interval into two parts, find both areas, and then add them? Or would I simply ignore the part of the...

Homework Statement Use the left endpoint graph with the given number of rectangles to approximate the area bounded by the curve f (x), the x-axis, and the line x = 4. f(x)=x2+x Homework Equations No idea. The Attempt at a Solution Once again, not a clue how to start this.

Homework Statement Attached question Homework Equations The Attempt at a Solution I tried rearranging S1 for Z then using Maple to plot it, which gave me a cone extending from the point z=1. For S2, would I have to plot it twice? once for <1 and once for =1? I have no...

Not really homework, but a textbook-style question... Homework Statement Is every subset of a totally bounded set (of a metric space) totally bounded? Homework Equations F is said to be totally bounded if, for every \epsilon>0, there's a finite subset F_0\subset F such that...

Hello--- I've been working on a problem which requires the numerical evaluation of an improper integral. I would like to transform the limits of integration on [0,\infty) to the bounded region [a,b] by replacing the variable \omega with another variable. Here is the integral...

Using polar coordinates to find the volume of a bounded solid[Solved] I found the equation of the boundary circle by setting z to 4 in the paraboloid. Then I did some work to get polar coords: x^2+y^2 = 1 x^2+y^2 = r^2 1-x^2-y^2 = 1-r^2 Then I set up my integral as such...

I asked by someone. But I can't answer to it. \sum^{\infty}_{1}(-1)^n*(1+\frac{1}{n})^n Is this series bounded? I can't do anything about that.

Homework Statement Part A) Establish which of the following combinations of particles can exist in a state of I=1 : a) \pi^0\pi^0 b) \pi^+\pi^- c) \pi^+\pi^+ d) \Sigma^0\pi^0 e) \Lambda^0\pi^0 Part B) of the problem is: In what states of isospin may exist the following systems? f)...

Homework Statement Bounded by the cylinders x2 + y2 = r2 and y2 + z2 = r2 We're supposed to stick to double integrals as triple integrals are taught in a later section. The Attempt at a Solution Edit: Alright, I think I go to the right answer. x = sqrt(r2 - y2) z = sqrt(r[SUP2[/SUP] - y2)...

Is a bounded set synonymous to a set that goes to infinity? I feel like unless a set is (-infinity, n) or [n, infinity) it is not going to be unbounded. The other thing that I was wondering is can a set be neither open nor closed AND unbounded? Doesn't the definition of open/closed imply...

Homework Statement Hey, the original question is not in english, so I am translating. So just to make sure I'm understood, i take convex to mean that the graph of the function is below the tangent. The question: Let F be a convex function and F is bounded from above by some number C, prove...

Homework Statement Let x_m = 1 + \frac{1}{2} + \frac{1}{3} + ... \frac{1}{m}, m \in N. Prove x_m is not bounded above and therefore x_m does not converge.Homework Equations We know from our class an important theorem stating that: If sequence converges then the sequence is bounded. Thus we...

hi. i'm reading "quantum mechanics in hilbert space" and a don't get a basic point for bounded operators. def. 1 a set S in a normed space N is bounded if there is a constant C such that \left\| f \right\| \leq C ~~~~~ \forall f \in S def. 2 a transformation is called bounded if it maps...

Homework Statement If f(t) transforms into F(s), so that \[ F(s) = \frac{{s + 1}}{{s^2 + as + 1}},a \in \] , prove that if a < 0, the function f(t) isn't bounded, and if a >= 0, it is bounded. Prove that if -2 < a < 2, f(t) oscilates. The Attempt at a Solution I honestly have...

Homework Statement Assume the theorem that a continuous bounded function on a closed interval is bounded and attains its bounds. Prove that if f: R -> R is continuous and tends to +\infty as x tends to +/- \infty then there exists an x0 in R such that f(x) \geq f(x0) for all x in R...

Homework Statement E1 = {pn(t) = nt(1-t)n:n in N}; E2 = {pn(t) = t + (1/2)t2 +...+(1/n)tn: n in N}; where N is set of natural numbers is the polynomial bounded w.r.t the supremum norm on P[0,1]? Homework Equations supremum norm = ||*|| = sup{|pn(t)|: t in [0,1]} The Attempt...

Homework Statement Find the area bounded between the two curves y=34ln(x) and y=xln(x) Homework Equations Integration by parts: \intudv= uv-\intvdu The Attempt at a Solution First I found the intersection points of the two equation to set the upper and lower bounds. The lower...

Homework Statement f is of bounded variation on [a;b] if there exist a number K such that \sum^{n}_{k=1}|f(ak)-f(ak-1)| \leqK a=a_0<a_1<...<a_n=b; the smallest K is the total variation of f I need to prove that 2) if f is of bounded variation on [a;b], then it is integrable on [a;b] 2...

hi, i have a hard problem, i guess so, i am looking for any help g(x) is a bounded Lebesgue measurable function that is periodic i.e. g(x)=g(x+p). Then for every f \in L^1(\Re) lim_{n\rightarrow \infty}\int_{\Re}f(x)g(nx) dx=(\int_{\Re}f(x)dx)((1/p){\int_{0}^{p}g(x) dx) thanks for...

Homework Statement f is of bounded variation on [a;b] if there exist a number K such that \sum^{n}_{k=1}|f(ak)-f(ak-1)| \leqK a=a_0<a_1<...<a_n=b; the smallest K is the total variation of f I need to prove that 1) if f is of bounded variation on [a;b] then it is bounded on [a;b]...

In many statements in probability, there is an assumption like bounded fourth moment, so is there any random variable which has unbounded fourth moment?

Homework Statement Given this ode system: x' = 2x+y-7e^(-t) -3 y'= -x+2y-1 Find all the bounded soloution in [a,infinity) when a is a real number... I'm not really sure what is a sufficient condition for bounded soloution in this question...Maybe there's something we can do and then we...

Hello, Just reading an essay about spherical harmonics and it says that spherical harmonic form a complete orthonormal basis set of functions over the sphere and can be used to represent any bounded single-valued function over a sphere. I am not sure I understand why we can only represent...

In general, would it be true that if a set is bounded, there must also be a supremum for the set? Too obvious, perhaps?

Homework Statement Find area of region bounded by curve with equation y=e^2x , the x-axis and the lines x=-ln3 and x=-ln2. Homework Equations log. law + integration The Attempt at a Solution well here is how i started this: y=e^2x after integration (1/2e^2x)...