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
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...
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...
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...
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!
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\\...
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...
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...
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...
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...
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...
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?
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...
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...
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...
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...
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...
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...
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)}...
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)...
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...
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...
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...
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...
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...
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 ...
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)...
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...
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?
[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...
[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...
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)...
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.
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...
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...
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 =...
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?
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?
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...
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...
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...
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...
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...
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?
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) |...
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...
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...
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...