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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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minimum or infimum for countable sets?

Suppose we have a countable set $X$, say $X=\mathbb{N}$, and let $Q \colon X \rightarrow \{0,1\}$ be a function. Is $$\min \{x \in X \colon Q(x)=1\}$$ the same as $$ \inf \{x \in X \colon Q(x)=1\}.$$ What about $\max$ and $\sup$? And for…
user136457
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If $\sup A = 5$ and $B = \left\{ 3a \mid a \in A \right\}$ then $\sup B = 15$

Prove that if $A \subset \mathbb{R}$, $\sup A = 5$, and $B = \left\{ 3a \mid a \in A \right\}$, then $\sup = 15$. I tried to do contradiction by assuming the hypothesis and that there is a number $< 15$ that is the supremum of $B$. Then $3a < 15$…
mr eyeglasses
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How to find infimum and supremum

I have to find the infimum and supremum of the set $A = \left\{\frac{n + k^2}{2^n + k^2 + 1} : n,k \in \mathbb{N}\right\}$. We assume $0 \notin \mathbb{N}$. $\inf A = 0$ because $\lim_{n \to \infty}{\frac{n + k^2}{2^n + k^2 + 1}} = 0$ but I don't…
alex
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Certain property of supremum

Given real numbers $x_{ni}, n\in \mathbb{N}, i\in I$, does it hold that $$\sup \bigg\{ \sum_{n\in\mathbb{N}}x_{ni}|i\in I\bigg\}=\sum_{n\in \mathbb{N}}\sup\bigg\{x_{ni}|i\in I \bigg\}?$$ Thanks in advance.
Greg P.
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Infimum of pre-image of continuous function

Let $f$ be a continuous function on $\mathbb{R}_+$ into $\mathbb{R}$. Then for every $n\in\mathbb{N}$ $$\inf\{t\in\mathbb{R}_+\colon f(t)\in [n,\infty)\}=\inf\{t\in\mathbb{R}_+\colon f(t)\in (n,\infty)\}\;.$$ Looking at the interval $[n,\infty)$…
Cinnamon
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The supremum of a set $A$

Does there exist a natural number $n$ for which there is a supremum of a set $$A = \{a\in\mathbb{Q}^+ | a^3+a\leq n^2\}$$ in the set of rational numbers? Since $f(a) = a^3+a$ is increasing function for $a\in \mathbb{Q}^+$, set $A$ has upper bound…
abelian25
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What is the reasoning behind this rigorous proof of the infimum of $A$?

Take the set $$A = \left\{\frac{n^2-1}{n^3-1} : n \in \mathbb{N}, n \neq 1 \right\}.$$ I'm proving that $\inf A = 0$. I've already shown the lower bound is $0$, so the only thing left to do is show any other lower bound of $A$ is less than or equal…
Bill Cogn
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I found the argument for why $\sup(a,b) = b$ in my textbook unconvincing. Please advice.

Here's the proof for why $\sup(a,b) = b$ in AOPS Calculus: $x < b$ for any $x ∈ (a, b)$, so $b$ is an upper bound. $x ∈ (a, b)$ for any $x ∈ \Bbb R$ and $a < x < b$, so there is no upper bound less than $b$ $⟹ \sup(a, b) ≥ b \quad(1)$ But $\sup(a,b)…
ten_to_tenth
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Min/max, inf/sup

Determine if the sets: (a){x ∈ Q} ∩ {${1 \over \sqrt{n}}$ : n ∈ N} (b) {x ∈ R : $x^2 −2>0$ } Have a minimum, a maximum, an infimum, and a supremum. For a: If n = 1, ${1 \over \sqrt{n}} = 1 $ and it's a maximum, 1 is also a supremum. There is't…
I don't need a name
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Infimum of $\left\{\frac{1}{\sqrt[n]{m}} + \frac{1}{\sqrt[m]{n}}, n, m \in \mathbb{N}\right\}$

I am trying to find infimum of $\Big\{\frac{1}{\sqrt[n]{m}} + \frac{1}{\sqrt[m]{n}}, \quad n, m \in \mathbb{N}\Big\}$ First, by using geometric mean $\leq$ arithmetic mean, I get: $$\sqrt[n]{m} = \sqrt[n]{m \cdot 1^{n-1}} \leq \frac{m + (n - 1)}{n}…
Snake Detection Hypothesis
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Supremum of an indicator function

I'm having trouble with the following. I have the test statistic $S_n(x)$: $S_n(x)=\sup_{x\in R}|Q_n(x)|$, where $Q_n(x)=1/\sqrt{n}\sum_{t=1}^n\triangle y_t 1\{y_{t-1}\leq x\}$, where $1\{\}$ is an indicator function and $\triangle y_t=y_t-y_{t-1}$.…
Søren Thode
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Supremum and infimum being the endpoints

Say we have the set $[1,5]$, this being a closed set, we know the infimum of supremum of this set is inside the set. Shouldn't the supremum in this case be $5$ and the infimum be $1$? Also in general, in a closed set [a,b], can we say supremum and…
Ellie_Wong
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Supremum of $\{1+\frac{2}{n}:n\in\mathbb{N}\}$

Problem. Let $A = \{1 + \frac{2}{n} : n \text{ is a natural number} \}$. Find $\sup A$, with justification. Isn't it just $3$ because $n$ is a natural number, and the lowest natural number is $1$, so the sup of this set should be $1+\frac{2}{1} =…
Amogus Impostor
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Is the supremum a function of a function just the maximum

Is $$ \sup_{x\in S} f(x) \equiv \max_{x\in S} f(x) $$ And is the correct definition of a maximum of a function the following: $$ \max_{x\in S} f(x):= y\quad \text{such that}\ \exists x\in S,y=f(x)\wedge\forall x\in S, f(x)\le y $$
Ben
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Understanding Archimedean Priniciple Proof in WikiProof

I am trying to understand the proof of archimedean priniciple stated on wiki proof here.(https://proofwiki.org/wiki/Archimedean_Principle). I am having trouble understanding the last part of the proof where it proves there is a supremum $$s =…
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