Supremum Definition and 138 Threads

Homework Statement http://gyazo.com/d59c730eb9b18dda4504a5fe118c7213 Homework Equations Limit and supremum. The Attempt at a Solution (a) Let : ##b_n = a_n - b## so that ##b_n ≤ 0## Now, ##lim(b_n) = lim(a_n - b) ≤ 0 \Rightarrow a - b ≤ 0 \Rightarrow a ≤ b## Q.E.D (b)...

Homework Statement For each subset of ℝ, give its supremum and its maximum. Justify the answer. {r \in \mathbb{Q} : r2 ≤ 5} Homework Equations Maximum: If an upper bound m for S is a member of S, then m is called the maximum. Supremum: Let S be a nonempty set of ℝ. If S is...

Homework Statement here's the picture and it's the second part of question 5: http://imgur.com/ybSW4v4 Homework Equations N/A The Attempt at a Solution so by intuition, I suspect that b = sup{a_n: n is in the natural numbers} If we can show that, then it will follow from...

Given : 1) f : [a,b] => R 2) f is continuous over [a,b] 3) f(a)<d<f(b) 4) a<b Then prove that the following,set: H ={x: xε(a,b),f(x)<d} has a supremum

Homework Statement Let $$S = \left\{ {\frac{n}{{n + m}}:n,m \in N} \right\}$$. Prove that sup S =1 and inf S = 0 Homework Equations The Attempt at a Solution So I was given the fact that for an upper bound u to become the supremum of a set S, for every ε>0 there is $$x \in S$$ such that x>u-ε...

Hello everyone! I'm really stuck on this one, it looks so obvious, but I can't prove it: Let $\alpha = \sup _{x\in [a,b]} f(x)$, how can I show that $\alpha = \sup _{x\in [a+c,b+c]} f(x+c)$? Thanks!

The following is my book's proof that $\sup\left\{x\in\mathbb{Q}:x>0, ~ x^2<2\right\} = \sqrt{2}.$ http://www.mathhelpboards.com/attachment.php?attachmentid=527&d=1356865161 I don't follow the bit where it says "if s were irrational, then w = \frac{\lfloor(n+1)s\rfloor}{n+1}+\frac{1}{n+1}."...

formalize the following definition: We define the supremum of a non empty subset of the real Nos (S) bounded from above ,denoted by Sup(S), to be a real No a ,which is the smallest of all its upper bounds

Hello everyone, I found this exercise on the internet: find the supremum and infimum of the following set A1, where A1 = {2(-1)^(n+1)+(-1)^((n^2+n)/2)(2+3/n): n belongs to |N*} being |N* = |N\{0} The solution was: A1 = {-3, -11/2, 5}U{3/4k, -3/(4k+1),-4-3/(4k+2),4+3/(4k+3) : k belongs...

Hello everyone! Given a set A that has a supremum $\alpha$, I want to show that $\alpha \notin int(A)$. Is the following proof accepted? $\alpha = \sup A$ so $\alpha$ is a limit point of $A$. If $\alpha \notin A$, we are done. Otherwise, for $\forall r>0$, we have $N(\alpha,r)-\{\alpha\}...

i was trying to formalize the definition of the supremum in the real Nos (supremum is the least upper bound that a non empty set of the real Nos bounded from above has ) but the least upper part got me stuck. Can anybody help?

Say we have a = \sup \{ a_{1}, a_{2}, a_{3}, ... \}. Then does this mean we can find some a_{n} \in \{ a_{1}, a_{2}, ... \} such that |a - a_{n}| < \varepsilon ? My reasoning is that since a (the supremum) is the least upper bound of the set, we have to be able to find some member of the set...

Prove that a nonempty finite S\,\subseteq\,\mathbb{R} contains its Supremum. If S is a finite subset of ℝ less than or equal to ℝ, then ∃ a value "t" belonging to S such that t ≥ s where s ∈ S. This is the only way I see to prove it, I hope your help :)) Regards

Are the least upper bound and supremum of a ordered field same thing? If so, then why do we have two different terms and why do textbooks do not use them interchangeably. That also means that greatest lower bound and infimum are also the same thing.

Homework Statement For each subset of ℝ, give its supremum and maximum, if they exist. Otherwise, write none. Homework Equations d) (0,4) The Attempt at a Solution For part d, if the problem were [0,4], both the supremum and maximum would be 4, since the interval includes the end...

$S = \{x : (x - a)(x - b)(x - c)(x - d) < 0\}$, where $a < b < c < d$ This questioned shouldn't be to difficult but would it be best to multiply out? And how is the $a < b < c < d$ going to affect it?

Homework Statement Prove that for every a in ℝ, and the set E = {rεQ : r<a} the following equality holds: a = sup(E) The Attempt at a Solution I'm not sure where to go? should i do this by contradiction that a≠sup(e) or should i do a traditional supremum proof or should i even do an epsilon...

In class, we have been introduced to the supremum and infimum concepts and shown them on graphs, but I am wondering how to go about deriving them, and determining if they are part of the set, without actually having to graph them- especially for more complicated sets.

Homework Statement Find the Supremum and Infimum of S where, S = {(1/2n) : n is an integer, but not including 0} Homework Equations The Attempt at a Solution Is it right if I got inf{S} = -∞ and sup{S} = ∞

Homework Statement In usual textbooks, what are the meanings of the notations (1) \sup_{k\in\mathbb{N}}|x_{k}| (2) \sup_{x\in [ a , b ] }|f(x)-g(x)| where (x_{k}) is a real sequence and f and g are real valued functions Homework Equations None. The Attempt at a...

(a) Let $S$ be a bounded non-empty subset of $\mathbb{R}$, and $\overline{m}=\sup S$. Prove there is a sequence $\{a_n\}$ such that $a_n\in S$ for all $n$, and $a_n\to\overline{m}$. (You must show how to construct the sequence $a_n$.) (b) Let $A$ and $B$ be bounded non-empty subsets of...

Let $S$ and $T$ be non-empty subsets of $\mathbb{R}$, and suppose that for all $s\in S$ and $t\in T$, we have $s\le t$. Prove that $\sup S\le\inf T$.

If $c>0$, prove that $\sup cA=c\sup A$ and $\inf cA=c\inf A$ My proof: $x\le\sup A$ for all $x\in A$. $cx\le c\sup A$ for all $x\in A$ ie $x\le c\sup A$ for all $x\in cA$. ------ (1) $x\le b$ for all $x\in A\implies\sup A\le b$ $cx\le cb$ for all $x\in A\implies c\sup A\le cb$ $x\le cb$...

Hello everyone, is the following an equivalent definition of the supremum of a set M, M subset of R? y=sup{M} if and only if given that y is an upper bound of M and x is any real number, y >= x implies there exists m in M so that m >=x. pf: Let x_n be a sequence approaching y from...

Homework Statement What is \displaystyle \sup_{\substack{x\in [-1.7,1.4] \\ y\in\mathbb{R} }} \frac{2.6xy}{(y^2+1)^2} (the supremum of \displaystyle \frac{2.6xy}{(y^2+1)^2} over x\in [-1.7,1.4] and y\in\mathbb{R})? The Attempt at a Solution How do I find the least upper bound?

Let {fi}i E I be a family of real-valued functions Rn->R. Define a function f(x) =sup fi(x) i E I[/color] 1) I'm having some trouble understanding what the sup over i E I of a function of x means? The usual "sup" that I've seen is something like supf(x) x E S for some set S. But...

Folks, Could anyone give me a working example of a sequence of functions that converges to a function wrt to the supremum norm? Thank you.

thanks!

Hey guys! Revising for an exam and I've come across a pretty basic problem. Question: Prove that the supremum of the set A : { 3n / (5n+1) :n€N} is 3/5 My answer: So 3n / (5n+1) ≤ 3n / 5n = 3/5 so 3/5 is an upper bound. Now, We claim that 3/5 is the least upper bound. Take β < 3/5...

Hello! I had a test in which the question that I will present here was asked. I got no points for my attempt at a solution. Do you think that I was still on the right track and that I deserve partial points? Here is the question: "A number M is said to be an upper bound to a set A if M...

So I've got a calculus test in a week, and I'm studying for it but I can't understand some examples our professor has given us. So, he says: 1) A = { x\in ℝ: (x-a)(x-b)(x-c) < 0 } , a<b<c. Find the supA and infA. In the solution of his example he says. It is easy to see that A =...

Homework Statement Find the supremum of E=(0,1) Homework Equations The Attempt at a Solution By definition of open interval, x<1 for all x in E. So 1 is an upper bound. Let M be any upper bound. We must show 1<=M. Can I just say that any upper bound of M must be greater than or...

Homework Statement Suppose a = sup(A) and b = sup(B). Let A + B = \left\{x + y;x\inA; y\inB\right\}. Show that a + b = sup(A + B). Homework Equations The Attempt at a Solution I'm honestly not sure where to start. Any help guys?

My question involves supremums and their implications: say I have the sequences \left\{x_{k}\right\}_{k=1}^{\infty} and \left\{y_{k}\right\}_{k=1}^{\infty} and I know sup \left\{x_{k}:k\in N \right\} \leq sup \left\{y_{k}:k\in N \right\} What can I say about the sequence...

Hi, I was wondering if I correctly applied the properties of the supremum of a set to solve the following proof. I feel like I "cheated" in the sense that I said, "Let s = Sup(B) - epsilon. Homework Statement If \sup A < \sup B, then show that there exists an element b \in B that is an upper...

Homework Statement well, the problem asks me to find the supremum(lub) of the set A={2x+sqrt(2)y : 0<x<1 , -1<y<2}. It's easy to show that for any x and y given in the defined domain, we have: -sqrt(2) < 2x+sqrt(2)y< 1+2sqrt(2). well, from this inequality, It's clearly seen that 1+2sqrt(2) is...

Homework Statement Let A be a set of real numbers that is bounded above and let B be a subset of real numbers such that A (intersect) B is non-empty. Show that sup (A(intersect)B) <= sup A The Attempt at a Solution I don't know how to start but tried this... Let C = A (intersect) B So...

The problem is attached. So is the solution by contradiction. I would love to hear your feedback if my sol. by contradiction is sufficient enough. Thanks a lot.

Homework Statement The problem is attached. The official solution to this problem is a proof from the contrary. I decided to go the straight-forward way. Would you check if I am correct? Thank you in advance. The Attempt at a Solution

Homework Statement find the supremum, infinum, maximum and minimum Homework Equations (n-2sqrt(n)) n is element of natural numbers The Attempt at a Solution no idea on how to do this help please

Homework Statement Let S:={1-(-1)^n /n: n in N} Find supS.The Attempt at a Solution Is this the right way to write the solution? Thanks! First, I want to show that 2 is an upper bound. For any n in N, 1-(-1)^n /n is less or equal to 2. Thus, n=2 is an upper bound. Second, I want to show...

I know .999... = 1. I'm just arguing against this method of proof. A common proof I see that .999 \ldots = 1 is that sup\{.9, .99, .999, \ldots \} = 1, but this is only true if you assume .999 \ldots \ge 1. If you assume, as most argue, that .999 \ldots < 1, then sup \{.9, .99, .999, \ldots \}...

Hi everybody, Please help me to find supremum & infimum of the set of rational numbers between √2 to √3 (ie) sup & inf of {x/ √2 < x < √3 , where x is rational number}

I do not understand how to calculate the sup\Omega and inf\Omega of a subset of R. So for example calculating the sup and inf of \Omega = (1,7)U[8,\infty) and the answer is no sup and inf = 1. I do not know how to get these values?

If S=sup {Sn: n>N}, is it true that kS= sup{kSn: n>N} where k is any scalar? Thanks!

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...