What is Supremum: Definition and 144 Discussions

In mathematics, the infimum (abbreviated inf; plural infima) of a subset



S


{\displaystyle S}
of a partially ordered set



T


{\displaystyle T}
is the greatest element in



T


{\displaystyle T}
that is less than or equal to all elements of



S
,


{\displaystyle S,}
if such an element exists. Consequently, the term greatest lower bound (abbreviated as GLB) is also commonly used.The supremum (abbreviated sup; plural suprema) of a subset



S


{\displaystyle S}
of a partially ordered set



T


{\displaystyle T}
is the least element in



T


{\displaystyle T}
that is greater than or equal to all elements of



S
,


{\displaystyle S,}
if such an element exists. Consequently, the supremum is also referred to as the least upper bound (or LUB).The infimum is in a precise sense dual to the concept of a supremum. Infima and suprema of real numbers are common special cases that are important in analysis, and especially in Lebesgue integration. However, the general definitions remain valid in the more abstract setting of order theory where arbitrary partially ordered sets are considered.
The concepts of infimum and supremum are similar to minimum and maximum, but are more useful in analysis because they better characterize special sets which may have no minimum or maximum. For instance, the set of positive real numbers





R


+




{\displaystyle \mathbb {R} ^{+}}
(not including 0) does not have a minimum, because any given element of





R


+




{\displaystyle \mathbb {R} ^{+}}
could simply be divided in half resulting in a smaller number that is still in





R


+




{\displaystyle \mathbb {R} ^{+}}
. There is, however, exactly one infimum of the positive real numbers: 0, which is smaller than all the positive real numbers and greater than any other real number which could be used as a lower bound.

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  1. M

    What is the Supremum of a Set in ℝ?

    Homework Statement Let T be a set such that: T=\{t\in\mathbb{R}/t^{2}<2\} Homework Equations a) Justify the existence of a real number a such that a=Sup(T) b) Prove that the proposition a^{2}<2 is false. c) Suppose that a^{2}>2. Prove that we can find a contradiction with a=Sup(T). d)...
  2. S

    Is it possible to perform supremum with two parameters in the same function?

    Can I do this: sup_(x,y) f(x,y) = sup_y sup_x f(x,y) ?
  3. STEMucator

    What is the relationship between the limit and supremum of a sequence?

    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)...
  4. M

    Using The Completeness Axiom To Find Supremum and Saximum.

    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...
  5. K

    Can't do this supremum question

    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...
  6. S

    MHB .Proving Supremum of H: f(x)<d, [a,b]

    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
  7. D

    Supremum & Infimum Homework Statement

    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...
  8. O

    MHB Supremum of f(x) on [a,b] vs. supremum of f(x+c) on [a+c, b+c]

    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!
  9. M

    MHB Supremum Proof Concerning Sqrt[2]

    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}."...
  10. S

    MHB Supremum of a Non-Empty Subset of Real Nos

    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
  11. F

    Understanding Supremum and Infimum: A Brief Explanation

    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...
  12. O

    MHB The Supremum of a Set is not an Interior Point

    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\}...
  13. S

    What is the Definition of the Supremum in First Order Predicate Logic?

    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?
  14. H

    Question about supremum and infimum

    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...
  15. I

    Prove that a nonempty finite contains its Supremum

    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
  16. B

    Least Upper Bound and Supremum

    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.
  17. H

    Is the Maximum Always the Same as the Supremum in an Open Interval?

    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...
  18. K

    Show supremum of an interval made by a continuous, increasing function

    Homework Statement Let f be an increasing function defined on an open interval I and let c ϵ I. Suppose f is continuous at c. Prove sup{f(x)|x ϵ I and x < c} = f(c) Homework Equations The Attempt at a Solution Since I is an open interval and c is not able to be an end point...
  19. D

    MHB Supremum and Infimum of $S$: $a < b < c < d$

    $S = \{x : (x - a)(x - b)(x - c)(x - d) < 0\}$, 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?
  20. J

    Proving a is Supremum of Set E

    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...
  21. E

    How Do You Determine Supremum and Infimum Without Graphing?

    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.
  22. R

    Analysis: Infimum and Supremum

    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} = ∞
  23. J

    Questions on notations about supremum

    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...
  24. W

    Showing that a supremum is a maximum

    Homework Statement Let f be a positive integrable continuous function on ℝ. Fix a measurable set E such that E \subset [0,1]. Let s = sup_{β \subset ℝ} [\int_{E} f_{β}(x)dx] where f_{β}(x) = f(x + β). Show that s is actually a maximum (not just a supremum) that is, there is at...
  25. T

    Real Analysis Supremum of a Set Proof

    Homework Statement Let S be a non empty set that is bounded about and β = sup S. Prove that for each ε > 0 there exists a point x in S such that x > β - ε.Homework Equations The Attempt at a Solution I don't really know how to begin this. I know it's true; I'm looking at the problem and I'm...
  26. alexmahone

    MHB Proofs of Supremum Problem in $\mathbb{R}$

    (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...
  27. alexmahone

    MHB Another supremum and infimum problem

    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 $\sup S\le\inf T$.
  28. alexmahone

    MHB How Does Scaling a Set Affect Its Supremum and Infimum?

    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 $x\in A\implies c\sup A\le cb$ $x\le cb$...
  29. S

    Equivalent definition of the supremum

    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...
  30. T

    Finding the Supremum: Solving for the Least Upper Bound of a Function

    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?
  31. K

    Understanding Sup of Function fi(x) i E I

    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 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 they...
  32. B

    A working example wrt the supremum norm

    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.
  33. R

    Uniform Continuity and Supremum

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  34. T

    Basic problem with supremum question.

    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...
  35. C

    Solving Supremum Question: Is 4 the Right Answer?

    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...
  36. T

    Problem with supremum and infimum examples

    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 =...
  37. D

    Intro to Real Analysis: Supremum

    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...
  38. S

    Having a difficult time with supremum proof

    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?
  39. Somefantastik

    What is the definition of supremum and how can it be used to compare sequences?

    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...
  40. J

    Am I correctly using the properties of the supremum in this proof?

    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...
  41. A

    How to prove that something is the supremum of a set?

    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...
  42. L

    Solving Supremum of Sets Homework Statement

    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...
  43. K

    A problem on infimum and supremum. Prove.

    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.
  44. K

    A problem on infimum and supremum.

    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
  45. J

    Find the supremum, infinum, maximum and minimum

    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
  46. K

    Finding the Supremum of S: A Proof

    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...
  47. T

    Is the Supremum Proof of 0.999... = 1 Flawed?

    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 \}...
  48. R

    Supremum & infimum of the set of all rational numbers

    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}
  49. R

    Supremum and Infimum of a subset of R

    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?
  50. S

    Does Scalar Multiplication Preserve the Supremum in Simple Sets?

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