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Questions tagged [supremum-and-infimum]

For questions on suprema and infima. Use together with a subject area tag, such as (real-analysis) or (order-theory).

The supremum (plural suprema) of a subset $S$ of a partially ordered set $T$ is the least element of $T$ that is greater than or equal to all elements of $S$. It is usually denoted $\sup S$. The term least upper bound (abbreviated as lub or LUB) is also commonly used.

The infimum (plural infima) of a subset $S$ of a partially ordered set $T$ is the greatest element of $T$ that is less than or equal to all elements of $S$. It is usually denoted $\inf S$. The term greatest lower bound (abbreviated as glb or GLB) is also commonly used.

Suprema and infima of sets of real numbers are common special cases that are especially important in analysis. However, the general definitions remain valid in the more abstract setting of order theory where arbitrary partially ordered sets are considered.

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supremum and infimum example

can anyone give me a good example of how to find supremum and infimum not lim sup and lim inf. I try to find some good example online but it keep given me lim sup and lim inf. I just need one good example of how to find sup and inf with step by…
jason
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Showing that $\sup\{|f(x)-f(y)|, x,y\in X\}= \sup \ f - \inf \ f$

I need to show that, for $f:X\to \mathbb{R}$ bounded, we have: $$\sup\{|f(x)-f(y)|, x,y\in X\}= \sup f - \inf f$$ Well, I know that $$\sup\{|f(x)-f(y)|, x,y\in X\}\ge |f(x)-f(y)|$$ but in what this helps? I really have no idea in how to prove this…
Guerlando OCs
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Prove that $\sup(A-B) = \sup(A) - \inf(B)$

$A-B = \{a-b: a\in A, b\in B\}$. Prove that $\sup(A-B) = \sup(A) - \inf(B)$ OK, let $x=\sup(A), y=\sup(B)$: $a\in A \implies a\leq x$ $b\in B \implies b\leq y$ $a+b\leq x+y$ is a upper bound Take $\varepsilon > 0$ and find $a,b$ s.t.: $a>x-\dfrac…
Marc
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Question about the completeness axiom

I have been looking online and on lecure notes and I have observed that there are 2 definitions for the completeness axiom and I cannot relate them together. These are: 1) Every non-empty set of real numbers that is bounded above has a supremum.…
Valentin
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Proof of $\sup ST = \sup S \sup T$

Claim Let $S, T\subset$ ordered field $F$ Let $P_F=\{a\in F: a \gt 0\}$ Let $ST = \{st \in P_F\mid s \in S, t\in T\}$ Then, $\forall S,T \in P_F\;\;\;$$\sup ST = \sup S \sup T$ Proof Let $\sup S=\alpha,\; \sup T=\beta$ Then $\forall s \in S$ and…
snapper
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prove $(\sup{|f(x)|})^2\geq(\sup{f(x)})^2$

prove $(\sup{|f(x)|})^2\geq(\sup{f(x)})^2$. $|f(x)|\geq f(x)$ $\sup{|f(x)|}\geq\sup{f(x)}$ $(\sup{|f(x)|})^2\geq(\sup{f(x)})^2$ when $\sup{f(x)}\geq0$ ... I don't know how to prove the other half where $\sup{f(x)}<0$. Any ideas or new approaches?
user192013
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S = {x ∈ Q : x² < 2} Prove sup(S) = - inf(S)

Define $S = {x \in Q : x^2 < 2}$. Let $a = \sup(S)$ and $b = inf(S)$. Prove that $a = -b$. (without finding $a$ or $b$) I know $b ≤ x ≤ a \; ∀ x \in S$ and $∀ \epsilon > 0 \; ∃ y_1, y_2$ s.t. $y_1 > a - ϵ$ and $y_2 < b + ϵ$ But how do I prove $a =…
PPDS
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Comparing supremums of two sets

Let $X,Y⊆ℝ$ be two non-empty sets. Prove that if $\sup Y$ exists and $\forall x \in X \exists y \in Y$ s.t. $x \le y$, then $\sup X$ also exists and $\sup X \le \sup Y$. For this question, I proved $\sup X$ exists but I don't know how to show…
confused
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Infimum of the set $\{x\in \mathbb{Q}\;|\;x^2<2\}$?

My first year Analysis textbook at university includes examples to grasp the concepts of infimum, supremum, maximum, minimum, lower bound and upper bound in set theory for subsets of the real numbers $\mathbb{R}$. One of such examples is the…
Mew
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Proving sup(A + B) = sup A + sup B

There are two sets $A$ and $B$ which are bound and are not empty. Now we'll define another set as: $A + B = \{a + b | a \in A, b \in B\}.$ I need to prove that $\sup(A + B) = \sup A + \sup B.$ I don't know if the way in which I proved it is enough…
LaptopDestroyer
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Infimum $\frac{(n+1)^2}{2^n} $

I need to find infimum $\frac{(n+1)^2}{2^n}$. Candidate is $ 0$. The first part of infimum definition is clear. I don't understand how to find $N(\varepsilon)$: $$\frac{(n+1)^2}{2^n}<\varepsilon+0$$ Maybe I can make some upper estimate but my…
Okumo
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inf e sup of empty set?

If $S$ is an ordered set then the empty set is a subset of $S$. What are $\inf$ and $\sup$ of such set? To be honest I don't know what it could be. I'm not talking about of real numbers, but any ordered set. The definitions I have been given hold…
user8469759
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Find supremum and infimum of $A=\left\lbrace\frac{2013}{1+\epsilon+\epsilon^{-1}}:\epsilon\in(0,1)\right\rbrace$

Find $\sup{A},\inf{A},\max{A},\min{A}$ where: $$A=\left\lbrace\frac{2013}{1+\epsilon+\epsilon^{-1}}:\epsilon\in(0,1)\right\rbrace$$ I suspect that $\sup{A}=\frac{2013}{3}, \inf{A}=0$ and max and min don't exist, I can easily prove that my candidates…
qiubit
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Prove that infimum (A)=0 and that supremum (A)=1 in the following set

$$A=\{\frac{n}{m}:m,n \in \mathbb{Z}^+, m>n\}$$ Now, I know that, as $n$ approaches $0$ from above and as $m$ approaches infinity, $\frac{n}{m}$ gets arbitrarily close to $0$, but my professor doesn't accept waffly terms like 'arbitrarily close' as…
beep-boop
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Radius of a set intuition

The radius of a set $S \in \mathbb{R}^L$ is defined as: $$\operatorname{rad}(S) = \inf_{x \in \mathbb{R}^L} \sup_{y \in S} \|x-y\|$$ I interpret the definition as follows: the radius of $S$ is related to the smallest element in $\mathbb{R}^L$ (not…
user12632521
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