Supremum Definition and 144 Threads

Homework Statement Find sup{\epsilon| N\epsilon(X0 \subset S} for X0 = (1,2,-1,3); S = open 4-ball of radius 7 about (0,3,-2,2).Homework Equations If X1 is in Sr(X0) and |X - X1| < \epsilon = r - |X - X0| then X is in Sr(X0) The Attempt at a Solution This is my first foray into...

KG Binmore talks about Zeno's paradox of Achilles and the tortoise to motivate the idea of suprema for sets of real numbers: i.e. on what he calls the continuum property. But can't Achilles can catch the tortoise even without the continuum property, e.g. on a race track of rational...

Homework Statement We just started learning about supremums and infimums in my math proofs class. I am having trouble with the following question: Let x, y be real numbers with y - x > 1. Prove that there exists an integer n such that x < n < y. Hint--use the well ordering principle...

i proved that sin (1/x)<1/x prove that sup{xsin (1/x)|x>0}=1 if we say that A={xsin (1/x)|x>0} xsin (1/x)<x(1/x)=1 so one is upper bound now i need to prove that there is no smaller upper bound so that 1 is the supremum suppose that "t" is our smaller upper bound t<1 and...

Analysis , sequences, limits, supremum explanation needed :( So i have a question and the answer as well, but i will need some explanation. here is the Question Let S be a bounded nonempty subset of R and suppose supS ∉S . Prove that there is a nondecreasing sequence (Sn) of points in S such...

I have analysis quiz tomorrow and i am really poor at sequences. I don't know where to begin Let (sn) and (tn) be sequences in R. Assume that (sn) is bounded. Prove that liminf(sn +tn)≥liminfsn +liminftn, where we define −∞ + s = −∞ and +∞ + s = +∞ for any s ∈ R. -thanks

I have this analysis homework due tomorrow. This is one of my problems.Let (sn) and (tn) be sequences in R. Assume that lim sn = s ∈ R. Then lim sup(sn +tn) = s+limsup(tn).I don't even know how to approach it. Even though it seems very straight forward.

Prove the supremum exists :) Homework Statement Let A = {x:x in Q, x^3 < 2}. Prove that sup A exists. Guess the value of sup A. The Attempt at a Solution First we show that it is non-empty. We see that there is an element, 1 in the set, thus A is non-empty. Now we show that A is...

Find the supremum and infimum of S, where S is the set S = {√n − [√n] : n belongs to N} . Justify your claims. (Recall that if x belongs to R, then [x] := n where n is the largest integer less than or equal to x. For example, [7.6] = 7 and [8] = 8) ----I found my infimum to be 0 and...

Homework Statement Find the supremum and infimum of S, where S is the set S = {√n − [√n] : n belongs to N} . Justify your claims. (Recall that if x belongs to R, then [x] := n where n is the largest integer less than or equal to x. For example, [7.6] = 7 and [8] = 8) The Attempt at a...

If you're given a sequence \{x_n\}, do you have \sup_n x_n = \lim_{n\to \infty} \left( \max\limits_{1 \leq k \leq n} x_k \right) I've never seen this definition before, but it makes sense. ...and if it's NOT the same as the supremum...what *is* it?

Could anyone tell me where to find a proof of the fact that the supremumnorm is a norm? The supremum norm is also known as the uniform, Chebychev or the infinity norm.

Homework Statement Find two functions f, g \in C[0,1] (i.e. continuous functions on [0,1]) which do not satisfy 2 ||f||^2_{sup} + 2 ||g||^2_{sup} = ||f+g||^2_{sup} + ||f-g||^2_{sup} (where || \cdot ||_{sup} is the supremum or infinity norm) Homework Equations Parallelogram identity...

Homework Statement Find the infimum and supremum of each of the following sets; state whether the infimum and supremum belong to the set E. \item 1. ~~~~E={p/q \in \mathbf{Q} | p^2 < 5q^2 \mbox{ and } p,q >0}. \mbox{ Prove your result. } \item 2. ~~~~E={2-(-1)^n/n^2|n \in \mathbf{N}...

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...

f_n(x)=1,1\leq x\leq n\\ f_n(x)=0,1< n< \infty f_n converges to f which is 1 at the beggining f_n is 0 but when n goes to infinity its 1 so why sup(f_n(x)-f(x))=1 ? f is allways 1 but f_n is 0 and going to one in one case its 1-1 in the other its 0-1 the supremum is 0...

Homework Statement f(a) > c > f(b) A = { x : b > x > y > a implies f(a) > f(y) } let u = sup(A) show that f(u) = c Homework Equations I have no idea in particular, save for the definition of the supremum: \forall x \in A x \le u if v is an upper bound of A, then u \le v...

According to supremum and infimum principle, nonempty set A={x|x\inQ,x2<2} is upper bounded, so it should have a least upper bound. In fact, it dose not have least upper bound. Why? When the principle is valid?

Homework Statement Prove that if (i) \forall n\in N, u - (1/n) is not an upper bound of s (ii) \forall n\in N, u + (1/n) is an upper bound of S then, u = supS Homework Equations The Attempt at a Solution It (i) and (ii) are true, then \exists s\in S s.t. u - (1/n) < s and...

Is this what it is: "For every \epsilon > 0 there exists x\in A such that x \leq \inf A + \epsilon." ...and similarly for the supremum?

Homework Statement Consider the open interval (a,b). Prove that \mathrm{sup}{(a,b)} = b. Homework Equations N/A The Attempt at a Solution I'm terrible at these proofs so I would appreciate it if someone could verify (or correct) my solution. Proof: Clearly b is an upper bound...

Why do we define(by convention) that infimum of an empty set as \infty and supremum as -\infty?

i can't understand this part of the solution.. http://img6.imageshack.us/img6/4977/43245163.th.gif its seems like the same thing

Hello, I found the definition. If S is supremum of set A, then a) \forall x\in A:x\leq S b) \forall\varepsilon>0\;\exists x_0\in A:S-\varepsilon<x_0 Now let define set A=\{1,2,3,4,5\}. Is number 5 supremum of set A? Condition a) is satisfied, but b) is problem. If \varepsilon=0.1, there...

Hi, I'm trying to prove that b^{r+s}=b^r*b^s for any real r,s where b^r = sup{b^t:t \leq r} and t is rational. (This is prob 1.6f in Rudin) My question. Can one show that for two sets X and Y: sup(XY)=(supX)(supY) where XY = {x*y: x\in X, y\in Y} Thanks, E

Homework Statement Prove that the supremum is the least upper bound Homework Equations The Attempt at a Solution Proof: let x be an upper bound of a set S then x>=supS (by definition). If there exists an upper bound y and y<=SupS then y is not an upper bound (contradiction)...

Homework Statement Let S be a set of positive real numbers with an infimum c > 0 and let the set T = {\frac{1}{t} : t \in S}. Show that T has a supremum and what is it's value. The attempt at a solution Ok, so the value must be \frac{1}{c}. But I'm unsure how to start proving...

Homework Statement let Ai be a subset of the reals and i is element of I = (1,...,n) now let A = UNION i is element of I Ai show that sup(A) = supiEI(sup(A)) Homework Equations The Attempt at a Solution My idea of solving was taking the limits of both sides but I'm...

Statement to prove: (Note: Q is the set of all rational numbers) Let B = {r in Q: r > 0 and r² < 2} and α = sup B. Prove that α² = 2. My work on the proof: Let B = {r in Q: r > 0 and r² < 2} and α = sup B. Note 1 is in B so B is not empty. By definition of B, 0 is an upper bound of B...

Homework Statement Let a_1,a_2,\ldots and b_1,b_2,\ldots be bounded sequences of real numbers. Define the sets X, Y and Z as follows: \begin{align*} X &=\{x \in \mathbb{R} : a_n > x \text{ for infinitely many } n \} \\ Y &=\{y \in \mathbb{R} : b_n > y \text{ for infinitely many } n \} \\ Z...

Find the supremum and infimum of the following sets: \begin{array}{l} A = Q \cap (\sqrt 2 ,\infty ) \\ B = \{ n + \sin n|n \in Z^ + \} \\ C = \{ 0.1,0.01,0.001...\} \\ \end{array} From the definition of supremum, it is obvious that sup A does not exist, because for any...

Suppose that A,B \subseteq \Re^+ are non empty and bounded sets. Define + and . as the following set opetations: \begin{array}{l} A + B = \{ a + b|a \in A,b \in B\} \\ A.B = \{ ab|a \in A,b \in B\} \\ \end{array} Prove that \sup (A + B) \le \sup A + \sup B I started by letting a \in...

Does the empty set have a supremum ( least upper bound)? if yes, can anybody give me a proof please? if no, again a proof please?

Problem: Show an example of a sequence of measurable positive functions on (0,1) so that \left\|\underline{lim} f_{n}\right\| < \underline{lim}\left\|f_{n}\right\| for n\rightarrow\infty My work: I think its just the indicator function I_{[n,n+1]} Since \left\|\underline{lim}...

Let 0 < r < 1. Then \sup_{x\in[-r,r]}f(x)=f(r)}, right? However, the text I'm reading says it's f(-r). How could this be? For example, say r = 0.5, then the least upper bound of [-0.5, 0.5] is 0.5, or r, right? I don't see how it could be -r. Thanks for any help.

I am trying to prove the following. I have a solution below. Can you tell if I am on the right track. P.S. I am doing calculus after 14 yrs so I am very rusty and probably sound stupid 1- Let T be a non-empty subset of R. Assume T is bounded below. Consider the set S = -T = {-t|t is an...

let A,B be nonempty sets of real numbers, prove that: if A,B are bounded and they are disjoint, then supA doesn't equal supB. here's my proof: assume that supA=supB=c then for every a in A a<=c and for every b in B b<=c. bacuse A.B are bounded then: for every e>0 there exists x in A such...

i need to show that when A is a subset of B and B is a subset of R (A B are non empty sets) then: infB<=infA<=supA<=SupB here's what i did: if infA is in A then infA is in B, and by defintion of inf, infB<=infA. if infA isn't in A then for every e>0 we choose, infA+e is in A and so infA is...

I am trying to prove some things for a HW problem. Can you guys tell me if the following logic looks ok. Let Un and Vn be bounded sequences of real numbers. If Un<=Vn for every n , show that lim sup n--->infinity Un <= lim sup n--->infinity Vn. Here is what I wrote: Let E1 and E2 be...

let B = \{x\in\mathbb{R} : sinx \geq 0 \} find the supremum and infimum of this set. Ok well, since it is periodic I guess the point would be to note that the set will repeat ever 2\pi So then if we consider just between 0 and 2\pi supremum = \pi infimum = 0 if we consider all...

Hi. Can somebody please check my work, its this dumb proof in the textbook which is the most obvious thing. Let S and T be nonempty bounded subsets of R with S \subseteq T. Prove that inf T \leq inf S \leq sup S \leq sup T. I first broke it up into parts and tried to prove each part...

"Let A and B be bounded nontempty sets of real numbers. Let C={ab:a in A, b in B}. Prove that sup(C)=sup(A)sup(B)." Here's what I've done so far: By the completeness axiom/theorem A and B have suprema. Let sup(A)=z and sup(B)=y. For all e>0, there exists a in A and b in B such that z-e<a...

I've got a question in analysis: How to calculate the supremum of sin n for positive integers n? I have tried hard but still cannot figure out it. Thanks very much to answer my question in advance! :smile:

Hi, It has been awhile since I have taken calculus, and now I am in analysis. I need to know what is the difference between the infimum and minimum and what is the difference between supremum and maximum? I know there is a difference, I just don't understand how they could be. Thanks...