Supremum Definition and 138 Threads

  1. L

    I Supremum of a set, relations and order

    Hello, found this proof online, I was wondering why they defined r_2=r_1-(r_1^2-2)/(r_1+2)? i understand the numerator, because if i did r_1^2-4 then there might be a chance that this becomes negative. But for the denominator, instead of plus 2, can i do plus 10 as well? or some other number...
  2. M

    Can You Solve a Problem Using the Definition of Supremum?

    For this problem, My solution: Using definition of Supremum, (a) ##M ≥ s## for all s (b) ## K ≥ s## for all s implying ##K ≥ M## ##M ≥ s## ##M + \epsilon ≥ s + \epsilon## ##K ≥ s + \epsilon## (Defintion of upper bound) ##K ≥ M ≥ s + \epsilon## (b) in definition of Supremum ##M ≥ s +...
  3. P

    I Showing equivalence of two definitions of essential supremum

    Assume ##f: X\to\mathbb R## to be a measurable function on a measure space ##(X,\mathcal A,\mu)##. The first definition is ##\operatorname*{ess\,sup}\limits_X f=\inf A##, where $$A=\{a\in\mathbb R: \mu\{x\in X:f(x)>a\}=0\}$$ and the second is ##\operatorname*{ess\,sup}\limits_X f=\inf B## where...
  4. chwala

    Find the supremum of ##Y## if it exists. Justify your answer.

    Refreshing on old university notes...phew, not sure on this... Ok in my take, ##x>0##, and ##\dfrac{dy}{dx} = -3x^2=0, ⇒x=0## therefore, ##(x,y)=(0,\sqrt2)## is a critical point. Further, ##\dfrac{d^2y}{dx^2}(x=0)=-6x=-6⋅0=0, ⇒f(x)## has an inflection at ##(x,y)=(0,\sqrt2)##. The supremum of...
  5. L

    Proving the Infimum and Supremum: A Short Guide for Scientists

    Hi, I have problems with the proof for task a I started with the supremum first, but the proof for the infimum would go the same way. I used an epsilon neighborhood for the proof I then argued as follows that for ##b- \epsilon## the following holds ##b- \epsilon < b## and ##b- \epsilon \in...
  6. S

    On the definition of radius of convergence; a small supremum technicality

    I am reading the following passage in these lecture notes (chapter 10, in the proof of theorem 10.3) on power series (and have seen similar statements in other texts): I'm confused about ##|x_0|<R##. If ##M=\sup (A)##, then for every ##M'<M##, there exists an ##x\in A## such that ##x>M'##...
  7. J

    I Show ##sup\{a \in \mathbb{Q}: a^2 \leq 3\} = \sqrt{3}##

    I would wish to receive verification for my proof that ##sup\{a \in \mathbb{Q}: a^2 \leq 3\} = \sqrt{3}##. • It is easy to verify that ##A = \{a \in \mathbb{Q}: a^2 \leq 3\} \neq \varnothing##. For instance, ##1 \in \mathbb{Q}, 1^2 \leq 3## whence ##1 \in A##. • We claim that ##\sqrt{3}## is an...
  8. M

    My proof of the Geometry-Real Analysis theorem

    Consider a convex shape ##S## of positive area ##A## inside the unit square. Let ##a≤1## be the supremum of all subsets of the unit square that can be obtained as disjoint union of finitely many scaled and translated copies of ##S##. Partition the square into ##n×n## smaller squares (see...
  9. Mathvsphysics

    A Limits and Supremum: Is It True?

    We have ##a_n## converges in norm to ##a## and a set ##S## such that for all ##n\ge 0## $$\sup_{s\in S} <a_n,s><+\infty .$$ Is it true that ##\sup_{s\in S} <a,s><+\infty##
  10. S

    MHB Finding the Infimum and Supremum

    Hello, I feel like I am struggling with this more than I should. I can tell intuitively what the infimum and supremum are, but I am pretty sure that I need a more formal proof style answer. How would one actually prove this question?
  11. yucheng

    Contradictory Proof of Supremum of E: Is it Circular?

    N.B. I have inserted the proof here as reference. See the bolded text. My question is, isn't the reasoning "##x^{2}+5 \varepsilon<2,## thus ##(x+\varepsilon)^{2}<2 .## " circular? If we can already find an ##0<\varepsilon<1## such that ##x^{2}+5 \varepsilon<2,## Can't one also claim that " we...
  12. C

    I Supremum proof & relation to Universal quantifier

    In the following proof: I didn't understand the following part: Isn't it supposed to be : ## a > s_A - \epsilon >0 ## and ## b > s_B - \epsilon >0 ## Because to prove that ## s ## is a supremum, we need to prove the following: For every ## \epsilon > 0 ## there exists ## m \in M ## such...
  13. N

    Find the infimum and/or supremum and see if the set is bounded

    ##S_3 = \left\{ \ x∈ℝ : x^2+x+1≥0 \right\}## I am not sure if I have done this correctly. The infimum/supremum and maximum/minimum are confusing me a bit. This is how I started: ##x^2+x+1=0## ##x^2+x+ \frac1 4\ =\frac{-3} {4}\ ## ## \left\{ x^2+\frac 1 2\ \right\} ^2 +\frac 3 4\ = 0##...
  14. AutGuy98

    MHB Proof of an Infimum Being Equal to the Negative Form of a Supremum ()

    Hey guys, I'm kind of in a rush because I'll have to go to my classes soon here at USF Tampa, but I had one last problem for Intermediate Analysis that needs assistance. Thank you in advance to anyone providing it. Question being asked: "Let $A$ be a nonempty set of real numbers which is...
  15. AutGuy98

    MHB Infimum and Supremum of a Set (Need Help Finding Them)

    Hey guys, I have this Intermediate Analysis problem that I need help finding the answer to. This is what the question asks: "Find the supremum and infimum of each of the following sets (considered as subsets of the real numbers). If a supremum or infimum doesn’t exist, then say so. No formal...
  16. Eclair_de_XII

    I Is there anything wrong with how the supremum of a set is written?

    I'm just having random thoughts today, and I didn't know where to put this, since this isn't even a homework problem. Anyway, is my way of writing the supremum of a set correct syntax-wise, or no?
  17. Math Amateur

    I Application of Supremum Property .... Garling, Remarks on Theorem 3.1.1

    I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume I: Foundations and Elementary Real Analysis ... ... I am focused on Chapter 3: Convergent Sequences I need some help to fully understand some remarks by Garling made after the proof of Theorem 3.1.1...
  18. V

    Finding an upper bound that is not the supremum

    I just want to see if I did this correctly, the interval (0,1) has 2 as an upper bound but the supremum of S is 1. So M would be equal to 2? Thank you.
  19. J

    MHB Real Analysis - Convergence to Essential Supremum

    Problem: Let $\left(X, M, \mu\right)$ be a probability space. Suppose $f \in L^\infty\left(\mu\right)$ and $\left| \left| f \right| \right|_\infty > 0$. Prove that $lim_{n \rightarrow \infty} \frac{\int_{X}^{}\left| f \right|^{n+1} \,d\mu}{\int_{X}^{}\left| f \right|^{n} \,d\mu} = \left| \left|...
  20. NihalRi

    What is the proof for the limit superior?

    Homework Statement 2. Relevant equation Below is the definition of the limit superior The Attempt at a Solution I tried to start by considering two cases, case 1 in which the sequence does not converge and case 2 in which the sequence converges and got stuck with the second case. I know...
  21. evinda

    MHB Finding Supremum and Infimum of Sets with Inequalities

    Hello! (Wave) I want to find the supremum, infimum of the following sets: $$\{ x \in \mathbb{R}: 0<x^2-1<3\}, \{1+\frac{(-1)^n}{n}: n=1,2, \dots \}$$ For the first set I have thought the following: $$ 0<x^2-1<3 \Rightarrow 1<x^2<4 \Rightarrow x^2>1 \text{ and } x^2 <4 \Rightarrow (x>1 \text{...
  22. ertagon2

    MHB Sequences and their limits, convergence, supremum etc.

    Could someone check if my answers are right and help me with question 5?
  23. E

    MHB Supremum and Infimum of Bounded Sets Multiplication

    Hey all, I started to learn this subject, and i understtod how to find the supremum and infimum of a given set or function. but I have problem with one question which I can not solve, and I don't know how to start. This is the quesion: Given to bounded sets X and Y, which their element are REAL...
  24. Math Amateur

    MHB Supremum Property (AoC) .... etc .... Yet a further question/Issue ....

    I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with yet a further issue/problem with Theorem 2.1.45 concerning the Supremum Property (AoC), the Archimedean Property, and the...
  25. Math Amateur

    I Supremum Property (AoC) .... etc .... Yet a further question

    I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with yet a further issue/problem with Theorem 2.1.45 concerning the Supremum Property (AoC), the Archimedean Property, and the...
  26. Math Amateur

    MHB Supremum Property (AoC) .... etc .... Another question/Issue ....

    I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with another issue/problem with Theorem 2.1.45 concerning the Supremum Property (AoC), the Archimedean Property, and the Nested...
  27. Math Amateur

    I Supremum Property (AoC) .... etc .... Another question/Issue

    I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with another issue/problem with Theorem 2.1.45 concerning the Supremum Property (AoC), the Archimedean Property, and the Nested...
  28. Math Amateur

    MHB Supremum Property (AoC), Archimedean Property, Nested Intervals Theorem ....

    I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with Theorem 2.1.45 concerning the Supremum Property (AoC), the Archimedean Property, and the Nested Intervals Theorem ... ...
  29. Math Amateur

    I Supremum Property, Archimedean Property, Nested Intervals

    I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with Theorem 2.1.45 concerning the Supremum Property (AoC), the Archimedean Property, and the Nested Intervals Theorem ... ...
  30. Bunny-chan

    Supremum and infimum of specific sets

    Homework Statement I'm in need of some help to be able to determine the supremum and infimum of the following sets:A = \left\{ {mn\over 1+ m+n} \mid m, n \in \mathbb N \right\}B = \left\{ {mn\over 4m^2+m+n^2} \mid m, n \in \mathbb N \right\}C = \left\{ {m\over \vert m\vert +n} \mid m \in...
  31. i_hate_math

    I Upper bound and supremum problem

    Claim: Let A be a non-empty subset of R+ = {x ∈ R : x > 0} which is bounded above, and let B = {x2 : x ∈ A}. Then sup(B) = sup(A)2. a. Prove the claim. b. Does the claim still hold if we replace R+ with R? Explain briefly. So I have spent the past hours trying to prove this claim using the...
  32. mr_persistance

    Apostol's Calculus I. 3.12 - Verify Solution.

    1. If x and y are arbitrary real numbers with x < y, prove that there is at least one real z satisfying x<z<y.2. I'll be using this theorem: T 1.32 Let h be a given positive number and let S be a set of real numbers. (a) If S has a supremum, then for some x in S we have x > sup S - h.The Attempt...
  33. RJLiberator

    Infimum and Supremum, when they Do not exist in finite sets

    Homework Statement Give an example of each, or state that the request is impossible: 1) A finite set that contains its infimum, but not its supremum. 2) A bounded subset of ℚ that contains its supremum, but not its infimum. Homework EquationsThe Attempt at a Solution I either understand this...
  34. I

    Rational numbers, supremum (Is my proof correct?)

    Homework Statement Show that the set ##\{x \in \mathbf Q; x^2< 2 \}## has no least upper bound in ##\mathbf Q##; using that if ##r## were one then ##r^2=2##. Do this assuming that the real field haven't been constructed. Homework Equations N/A The Attempt at a Solution Attempt at proof: ##r\in...
  35. I

    Supremum, Infimum (Is my proof correct?)

    Homework Statement Let ##A## be a nonempty set of real numbers which is bounded below. Let ##-A## be the set of all numbers ##-x##, where ##x \in A##. Prove that ##\inf A = -\sup(-A)##. Homework Equations Definition: Suppose ##S## is an ordered set, ##E\subset S##, and ##E## is bounded above...
  36. W

    I How Do Supremum and Infimum Relate When s < t for All s in S and t in T?

    Let S and T be subsets of R such that s < t for each s ∈ S and each t ∈ T. Prove carefully that sup S ≤ inf T. Attempt: I start by using the definition for supremum and infinum, and let sup(S)= a and inf(T)= b i know that a> s and b< t for all s and t. How do i continue? , do i prove it...
  37. P

    I Supremum inside and outside a probability

    I'm trying to deal with the supremum concept in a specific situation, but I think I'm getting the concept wrong. A step of a proof I'm going through states: P\ [\sup\limits_{x}\ |f(x)\ -\ f'(x)|\ >\ y\ |\ z]\ \ \leq\ \ \sum_{i=1}^M\ P\ [\ |f(x)\ -\ f'(x)|\ >\ y\ |\ z]\ \ \leq\ \ M\times\...
  38. V

    Proof of points arbitrarily close to supremum

    Homework Statement Let S \subset \mathbb{R} be bounded above. Prove that s \in \mathbb{R} is the supremum of S iff. s is an upper bound of S and for all \epsilon > 0 , there exists x \in S such that |s - x| < \epsilon . Homework Equations **Assume I have only the basic proof...
  39. Alpharup

    B Doubt regarding least upper bound?

    I am using Spivak Calculus. I have a general doubt regarding the definition of least upper bound of sets. Let A be any set of real numbers and A is not a null set. Let S be the least upper bound of A. Then by definition "For every x belongs to A, x is lesser than or equal to S" Let M be an...
  40. P

    Supremum = least upper bound, anything > supremum?

    The supremum is defined as the "LEAST" upper bound. The word "least" makes me think, there is a "MOST" upper bound, or at least something bigger than a "least" upper bound. For a set of numbers, is there anything larger than a supremum? Supremum is analogous to a maximum, but I don't...
  41. P

    Am I understanding "supremum" correctly

    If let's say I have an expression: ##x\leq y## Since the supremum is defined as the "least upper bound," does this make sup(x) for this case ##x=y## or is it ##x = \infty##?
  42. Z

    Real Analysis - Infimum and Supremum Proof

    Hi Guys, I am self teaching myself analysis after a long period off. I have done the following proof but was hoping more experienced / adept mathematicians could look over it and see if it made sense. Homework Statement Question: Suppose D is a non empty set and that f: D → ℝ and g: D →ℝ. If...
  43. M

    Showing a sequence converges to its supremum

    Homework Statement : [/B]Let a = sup S. Show that there is a sequence x1, x2, ... ∈ S such that xn converges to a.Homework Equations : [/B]I know the definition of a supremum and convergence but how do I utilize these together?The Attempt at a Solution :[/B] Given a = sup S. We know that a =...
  44. O

    MHB Proof of Supremum of $M$ Mapping into Itself

    Let $f$ be a mapping of a metric space $M$ into itself. For $A\subset M$ let $\And(A)=sup\left\{d(a,b);a,b\in A\right\}$ and for each $x\in M$, let $O(x,n)=\left\{x,Tx,...{T}^{n}x\right\}$ $n=1,2,3...$ $O(x,\infty)=\left\{x,Tx,...\right\}$ Please prove that...
  45. O

    MHB Is Max A $\le$ Sup B for A $\subseteq$ B?

    for A$\subset$ B max A $\le$ sup B ? is it true ?
  46. S

    Infimum and supremum of empty set

    Hello, I can't wrap my mind around this: inf∅= ∞ sup∅= - ∞ Thank you in advance.
  47. R

    Rudin's Principles Theorem 1.11 (supremum, infimum)

    Mod note: Edited by removing [ sup ] tags. To the OP: Please don't fiddle with font tags, especially the SUP tag, which renders what you write in very small text (superscript). Hello everyone I have just started studying mathematics at university this summer and I have decided to supplement my...
  48. C

    How Do Transformations Affect the Supremum of a Set?

    Homework Statement 8. Let ##A## be a non-empty subset of ##R## which is bounded above. Define ##B = \{x ∈ R : x − 1 ∈ A\}##, ##C = \{x ∈ R : (x + 1)/2 ∈ A\}.## Prove that sup B = 1 + sup A, sup C = 2 sup A − 1. The attempt at a solution Note that ##sup A## exists. Let ##x ∈ B##; then ##x − 1...
  49. 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)...
  50. 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) ?
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