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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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Proving supremums of sequences

If $l$=sup($x_n$), what is sup($kx_n$) where k$\in$$\mathbb{R}^{+}$? Prove your conjecture. I have that sup($kx_n$)=$kl$. I can prove that it is an upper bound of $kx_n$, but I'm having trouble finishing the proof and showing that it is less than…
Lauralolo
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Absolute of a function equal to zero

How would I prove the following theorem: if $\sup \{ |f (x)| :x>0\} = 0$ this implies $|f (x)| = 0$ for all real number $x$.
Adnan
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basic question about infimum, minimum and countable/uncountable sets

I have not studied mathematics, so please be patient with me. Does it make sense to take the minimum over an uncountable set? In my opinion, if the set is closed in this minimum direction, then it seems to be valid to take the minimum. However, if…
user136457
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Min, Max, Infimum, Supremum

I am trying to understand the notion of minimum, maximum, infimum and supremum. Can you please comment on these solutions for the below examples? Minimum , Maximum, Infimum, Supremum : a.$(0,1)$ none, none, $0$, $…
vkoukou
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Findin the $\sup$ and $\inf$ of defined sets

Yet again struggling. Find $\sup A$ and $\inf A$ where $A$ is the set defined by: (a) $A=\{x∈\mathbb{Q}:x^{2} −x<1\}$ (b) $A=\{x∈\mathbb{R}:x^3 −x\le6\}$ My answers: (a) $\sup A= ?\quad\inf A=\frac12$ (b) $\sup A=2 \quad\inf A=-\infty$ Please…
Dumbanddumber
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Proving a complicated statement about supremum of a bounded set - how to proceed?

I'm trying to solve the following problem: Say whether this statement is true for every $A\subset\Bbb{R}$ bounded from above ($A\not=\varnothing$) and for every $d\in\Bbb{R}$: $$[(\sup{A} = d\notin A)\implies(\exists_{N\in\Bbb{N}}\forall_{n\ge N,…
qiubit
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checking the answer for infimum and supremum of a set

I want to find the infimum and supremum of the set $$S=\left\{\frac{3n+2}{2n+1}\mid n \in \mathbb N \right\}$$ I found $\inf S=\frac32$ and $\sup S=\frac53$. Is that correct?
manish
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Question about Infimum. (conclusion about intersection)

Let $A$ be an infinite set that includes Real numbers and is bounded. Let $B$ be a set of Real numbers $x$ s.t. the intersection $A\cap[x,\infty)$ is empty or includes $finite$ number of elements. prove that $\inf B$ exists. prove or disprove…
Firas Abd El Gani
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Determine whether the following subsets of $\mathbb{R}$ are bounded.

$A=\{x+\frac{1}{x}:x \in (0,\infty)\}$ $B=\{x^2+xy^2:-2 \leq x \leq 1, -1 \leq y \leq 1\}$ I understand the what it means for a set to be bounded above and below, but how would I go about proving this rigourously?
beep-boop
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Find the Infimum and supremum of M set

$$M = \bigcap_{n\in N}{I_n},$$ $$I_n = \bigcup_{k = 1}^{3^n - 2} \left(\frac{k}{3^n},\frac{k+1}{3^n}\right)$$
VadimAflin
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Proof to calculate supremum

I was wondering if someone could help me with a proof for the supremum of the set $E = \frac{1}{n} + (-1)^n$ where n is a natural number. I can see that the supremum is equal to 3/2 at n=1 but wasnt sure how to justify it. previously i have seen…
johnsdgh
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Prove that $m$ is a lower bound for $S$ if and only if $−m$ is an upper bound for $−S$.

Given : $−S = \{−s : s \in S\}.$ Prove that $m$ is a lower bound for $S$ if and only if $−m$ is an upper bound for $−S$. And prove that if $S$ is bounded below then its greatest lower bound satisfies $\inf S = − \sup(−S)$.
Aries
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How do I calculate the Supremum and Infimum of these sets

I am not sure how to approach these supremum and infimum sets. I also have to give a min or max $$A = \Big\{(-1)^n+\frac{1}{n+1}\ :\ n\in \Bbb{N}\Big\}$$ $$B = \Big\{\frac{x}{x+1}\ :\ x\in [0,\infty)\Big\}$$ $$C =…
MMoe
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Proof of Supremum of a set

Prove that $$\sup\{1−1/n^2\}=1.$$ I have tried to prove it by Archimedean property as let $a>1$ is the Supremum and tried to find contradiction but can't .
Mayank Kashyap
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What is the proof for sup{|x−y|, (x, y) ∈ A 2} = sup A−inf A?

So my question is about what is the proof that: $$\sup\{|x−y| : (x, y) \in A^2 \} = \sup A − \inf A$$ Please provide a detailed demonstration. Thanks in advance.
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