Mathematics Stack Exchange
Mathematics Stack Exchange

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.

2933 questions
1
vote
1 answer

to show supremum exists, in a set satisfying g. l. b property

I am asked to show if, $X$ be an ordered set with g.l.b property. Let $E\subseteq X$ be non empty and bdd. above. Prove that $\sup E$ exists in $X$. I tried like this: since $X$ has g.l.b property, so $\inf X$ exists , call it $l$, now since $E$ is…
Myshkin
  • 35,974
  • 27
  • 154
  • 332
1
vote
1 answer

Sets that don't have infima (infinum and supremum)

Scott domain is a non-empty partially ordered set if the following holds: D is bounded complete, i.e. all subsets of D that have some upper bound have a supremum. ... What would be an example of a set that doesn't have a supremum? I was under…
Stepan
  • 1,093
1
vote
3 answers

Why is the supremum this?

$M=\{y\in \mathbb R |y=3x+10:x\in(9,14)\}$ From what I've learned, this means $90$ there exists such $a\in A$ so $u-\epsilon
user1768788
  • 201
1
vote
1 answer

Supremum and infimum for xyz = 1

I have $A = \{x + y + z: x, y, z > 0, xyz = 1 \}$ and I'm investigating whether this set has an infimum and a supremum. It looks to me like there is no supremum as the set doesn't seem to be bounded from above. A candidate for an infimum seems to be…
Zelazny
  • 2,489
1
vote
0 answers

Supremum and infimum of set with n!

I want to conclude about $\sup A$ and $\inf A$ where $A = \{ {n^n \over (n!)^2}: n = 1, 2, \dots \}$. My intuition is that $\sup A = 1$ and $\inf A=0$. To show that the infimum exists, I could prove that ${n^n \over (n!)^2}$ can be "as small as we…
Zelazny
  • 2,489
1
vote
1 answer

How do you prove the infimum and supremum?

Suppose that $S$ is nonempty and bounded above. Show that the set: $$ -S:= \{-x \mid x \in S\} $$ is bounded below and that $\inf(-S) = -\sup(S)$.
MIKEEEE
  • 11
1
vote
0 answers

Supremum/Infimum if x in R^n

there is a function f(x)=(3+x^2)/(5+x^4). I should find the Supremum and Infimum BUT for x of R^n. I dont understand what x in R^n changes in this situation? I can see that the Supremum in this case would be f(x)=2/3 and the Infimum f(x)=0. I…
alitran
  • 19
1
vote
1 answer

Proof that if set A is contained in Set B, then the supremum of A is less than or equal to the supremum of B

If $A \subset B$, then $\sup_B \geq \sup_A$. $\textbf{Proof:}$ Let $k = \sup_B$. If $\sup_A$ were greater than $\sup_B$, then $\sup_A=k+\delta$ where $\delta$ is some positive number. Since, however, $k$ is the maximum value in $B$, $k+\delta$…
Izzy M.
  • 29
1
vote
1 answer

sup and inf of $\{\frac{(-1)^n}{n}, n\in \mathbb{N}\}$

I need to prove that $\sup A = \frac{1}{2}$ and $\inf A = -1$ for: $$A = \{\frac{(-1)^n}{n}, n\in \mathbb{N}\}$$ Well, for $n\in 2\mathbb{N}$, we have: $$A' = \{\frac{1}{n}, n\in 2\mathbb{N}\} = \{\frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \cdots\} =…
Guerlando OCs
  • 6,584
1
vote
2 answers

Give an example where the inequalities are strict $\sup (\inf A,\inf B) \le \inf (A \cap B) \le \sup (A \cap B) \le \inf (\sup A, \sup B)$

$A$ and $B$ are in the real set. $\sup (\inf A,\inf B) \le \inf (A \cap B) \le \sup (A \cap B) \le \inf (\sup A, \sup B)$ I proved the inequalties as I was asked to but couldn't find an example. Is it that I should take sets A and B as the inverse…
user315211
  • 59
1
vote
1 answer

Supremum and infimum - Are those examples right?

Hello did a few exercises about supermum and infimum but im not sure if my solutions are correct. The following set are given: (i) $\{n \text{ is element of the whole numbers } | n³ < 10\}$ Here was a little mistake. I think my answer is right now…
hprandom
  • 13
1
vote
2 answers

Find the supermum and infimum of a set

I Got the set $\sqrt[k]{k+1}$ I have found the supermum by the Inequality of arithmetic and geometric means. And the result is 2. I dont have a way for solving the infimum . I tried to solve it by move to to some eqaution and try by Binomial…
Shachar
  • 31
1
vote
1 answer

Find $\inf(X)$ and $\sup(X)$ then check by definition of supremum and infimum.

Yes, I also think that the question is strange, but I can't rephrase it. I need to find $\inf(X)$ and $\sup(X)$ if $X = \{X_n, X_n = 3 \sin 4n, n\in\mathbb{N}\}$. I'm sorry, but I don't even know how to start the process of finding the solution. If…
Mex
  • 91
1
vote
2 answers

Supremum Proof Question

Let $a
Laura
  • 171
1
vote
2 answers

Show that exists maximum of $A_x=\{m\in \mathbb{Z}: m\le x\} $

For every $x \in\mathbb{R}$, we define $A_x=\{m\in \mathbb{Z}: m\le x\} $. What i have done thus far is show that $A_x\neq\emptyset$, because if $x\in\mathbb{Z}, A_x=\{x\}$ and as $A_x\subset\mathbb{R}$ and it is bounded above by $x$, so by the…
Ariel Marcelo Pardo
  • 415
1 2 3
…
11 12