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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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Function definition involving supremum and infimum

I am currently reading Vector Calculus, Linear Algebra, and Differential Forms by John Hubbard and Barbara Hubbard and am having a bit of trouble reconciling notation with definition. The book is trying to express how to set up arithmetic for the…
OhLawdyLawdy
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Supremums and subsets

Let $A \subseteq B$, where $A$ and $B$ are non-empty sets and $B$ is bounded above. Show that $\sup(A)$ exists and $\sup(A)\le \sup(B)$ (Problem given as written) For clarity, $\sup(x)$ is the supremum of x I'm just asking if I did this correctly,…
user3400223
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Does maximum or supremum of an infinite set exit?

If I have a set: $S = \{ k_1, k_2,...,k_n,... : k_i < \infty \}$, which means that $S$ include infinite many elements, but each element is a finite real number. In this case does $\max(S)$ or $\sup(S)$ exit, and is finite? I can not find useful…
Amanda
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Am I right that this statement is false?

For every nonempty set $A$ of real numbers having an upper bound, and for every $d \in \mathbb{R}$, we examine the statement: $$[(\sup A = d, d \notin A) \implies (\exists_{N \in \mathbb{N}}{\forall_{n \ge N}{\exists_{a \in A}{\;d - \tfrac{1}{n} <…
matrix
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Supremum and infimum of an inequality

I feel like this is elementary but I can't seem to determine the supremum/infimum (if they exist) of the following example: Let $S = \{x \in \mathbb{R} : x^2 > 2x + 8\}$ I rewrote it as $$x^2 -2x - 8 > 0$$ $$(x-4)(x+2) > 0 $$ So it is clear that…
Pandamonium
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infimum and supremum finding

Find the $\sup$ and $\inf$ of $E=\{x\in\mathbb{R}\mid 1-\frac{1}{n} < x < 2-\frac{1}{n}, n \in \Bbb{N}\}$. Justify your answers I claim that $\sup E = 2$ and $\inf E = 0$. Let $(1-1/n,2-1/n]$, then $\sup E = 2-1/n$ and $\inf E = 1-1/n $. Case 1:…
user187296
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Infimum and supremum of the set of all numbers whose square is greater than 2

Suppose, $S= \{ x \in \mathbb R\mid x^2 > 2\}$. Then $\sup S = +\infty$. What is $\inf S$? I'm guessing that $\inf S \in (-\sqrt 2, \sqrt 2)$. Is that true?
Faithhhhhh
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an example of supremum using relation other than <. like using divides etc??

i want to know some examples of supremum or infimum on a poset using relation, other than $\le$, for example using divides or mod. basiclly i want to know how to find out the supremum of a poset for an arbitary relation. thank you
hari
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Existence of intervals $I_1$ and $I_2$ such that $\sup_{x \in I_{1}}f(x) \leq \inf_{x \in I_{2}}f(x)$ when f is contiuous

Suppose that $f:\mathbb{R} \rightarrow \mathbb{R}$ is a continuous function. Prove that there exist intervals $I_1$ and $I_2$ such that $\sup_{x \in I_{1}}f(x) \leq \inf_{x \in I_{2}}f(x)$ Now, we can consider any two intervals [a,b] and [c,d] so…
Tim
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Calculate the $\sup$

Let us $A=\{\frac{1}{|\frac{\pi}{2}+2n\pi|}, n\in\mathbb Z \}$, Calculate $\sup A$. I am confuse, because $n\in\mathbb Z$ then for me i need to find the infimum of $|\frac{\pi}{2}+2n\pi|$ that it goes to $\infty$ implies that $\sup A= 0$ right? I…
weymar andres
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Supremum of $\cos(n)/n$ for $n\geq 1$

How to find the supremum of $\{\cos(n)/n:n\in\mathbb{N}\}$? When I draw a graph of $x\mapsto \cos(x)/x$, it would be $\cos(1)$ that is the supremum. But, I can't prove it rigoriously, as it decreases and increases in some intervals due to…
Mr.MathDoctor
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Supremum of a set. Infinitely many upper bounds

Could anyone please help me figure out whether I understand this topic correctly? It is said that if a set has an upper bound, it may have infinitely many upper bounds. E.g. if we have a set $A = \{0,1\}$. Can I say that $2$ is an upper bound for…
Inna Khachikyan
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LUB and GLB of a function

Find the least upper bound (LUB) and greatest lower bound (GLB) of $\{x\sin(1/x):x>0\}$ My Attempt: Since limit of given function is $0$ as $\lim_{x\to f(x)}g(x) = 0$ when $f(x)\to0$ and $g(x)$ is bounded. So $\operatorname{LUB}(f) =…
Ankit
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Finding supremum & infimum

$B=a+(2a)^{-1}: a\in \mathbb Q, 0.1≤a≤5$. What I tried to do is work on the limits, $0.1≤a≤5 \implies 1/10≤1/2a≤5$ than to find the supremum and infimum. I tried to find the limit for $a+(2a)^{-1}$ which was $1/10≤a+1/2a≤5+1/10$. Making the…
Ria Park
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infimum and supremum of $(sin(a))^n and (cos(a))^n$

infimum and supremum of $(sin(a))^n$ and $(cos(a))^n$ with $a\in \mathbb{R}, n\in \mathbb{N}$ How do you show that $(sin(a))^n$ and $(cos(a))^n $ have supreme and infimum?
José Romero
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