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
0
votes
1 answer

What does the union and intersection of a single sequence of real number mean?

I read measure_theory by Paul R-Halmos, part of number 7 prerequisite concept of reading this book is: "The supremum and infimum of a sequence {$x_n$} of real numbers are denoted by $\bigcup_{i =1}^\infty x_i$ and $\bigcap_{i =1}^\infty x_i$." Why…
jimy
  • 1
  • 2
0
votes
1 answer

If $A$ is bounded then so is $A\cdot A$?

im struggling to prove the following problem. Let $A \subseteq \Bbb R$ be a non empty set, (multipication is for example $A\cdot B= \{a \cdot b\ |\ a \in A,\ b \in B\}$) Show: If $A$ is bounded, then $A\cdot A$ is bounded too! I know that from the…
Aviv Barel
  • 45
  • 5
0
votes
0 answers

Number of supremum(s) with n elements of set?

I´m trying to learn how to find the number of supremum(s) of a set with defined number of elements in it. Assume A = {1,2,3,4,5,6} and let R be relation defined on power set P(A), so 2 subsets X,Y of A are in relation if X is subset of Y. Now M =…
johnjohs
  • 1
0
votes
1 answer

Find supremum and infimum of a set

Find supremum and infimum of the sequence $$a_n = \frac{2n^2}{4n^2+1},\:\:n\:\in \mathbb{N}.$$ I thought the supremum is $1/2$ and there is no infimum because the sequence gets close to ${0}$, but $0\notin \mathbb{N}$. In addition, I need to find…
math112311
  • 41
0
votes
2 answers

How to prove the existence of infimum and supremum?

I have a set $A = \bigl\{\frac{n}{2^n} : n \in \Bbb N\bigr\}$. Now I want to prove that $0$ is the $\inf A$ and $\frac{1}{2}$ the $\sup A$. I'm doing it this way: I use the definition: $\sup(A)$ exists only if: 1)$\forall a\in A,~a\leq s$ 2)…
Elise Svart
  • 179
0
votes
1 answer

Find and then prove the supremum of set X

Set X is defined with $$\sqrt {(x (x + 1))}/(2 x + 1)$$, x > 0. I can't prove that for every epsilon greater than zero there exists an t > 0 such that $$1/2 - \sqrt{(t (t + 1))}/(2 t + 1) < \epsilon.$$ Every other step I get.
user680838
  • 11
0
votes
1 answer

How to prove that infimum of $A = \{\frac{n}{2^n} : n \in N\}$ is 0?

How to prove that infimum of $A = \{\frac{n}{2^n} : n \in N\}$ is 0? Could you explain this to me step by step?
Elise Svart
  • 179
0
votes
2 answers

If $||x|| < \sup\{||x_i||\}$ and $||x|| > \inf\{||x_i||\}$, then is $x \in \{x_i\}$?

For any set $X \subset \mathbb{R}^d$, is it true that if $$||x|| < \sup\{||x_i|| : x_i \in X\} \quad\text{ and } \quad ||x|| > \inf\{||x_i|| : x_i \in X\}$$ then $$x \in X?$$
on-pasta
  • 531
0
votes
1 answer

What does notation $\inf_{k,l}$ mean for indices $k,l$?

What does notation $\inf_{k,l}$ mean for indices $k,l$? Does it mean that one picks $\inf k$ and $\inf l$ or that one picks some kind of "inf of both $k$ and $l$" (whatever that means)?
mavavilj
  • 7,270
0
votes
0 answers

Infimum depending on two variables

How can I calculate the supremum of the infimum of a function depending on two variables? For example: $$ \begin{align*} a =\sup_{t \geq 0} \left\{ \inf_{r \geq 0} \{ 10 \leq t + r \} \right\} \end{align*} $$ What I would have done is…
maax
  • 101
0
votes
1 answer

mathematical conversion

I want to understand, how i get from the left to the right side in the following inequaltiy: $$\sup\left\{\lvert f(y_1)-f(y_2)\rvert : \lVert y_i-y\rVert<\nu\right\}\leq \sup\left\{\lvert f(y_1)-f(y_2)\rvert : \lVert y_i-x\rVert<\delta\right\}$$ I…
Leon1998
  • 401
0
votes
1 answer

Is this an equivalent way of writing supremum

Let $\{A_n\}$ be a sequence of events I have an indicator r.v. $I_n$ where $I_n=1$ if an event $A_n$ occurs and $0$ otherwise Let $X_n = \sum_{k=1}^n I_k$ Is it the same thing to say this $\sup_{n\in \{1,2,3...\}} E(X_n)$ is equal…
user130306
  • 1,890
0
votes
1 answer

Showing infimum of a set is smaller or equal to the infimum of a different set

Let $R= [a,b] \times [c,d]$ be an arbitrary rectangle. Define $S_1 =\{f(x,y): (x,y)\in R\} $ and $S_2=\{f(x,y):a\leq x \leq b\}$ Claim: For every $c\leq y \leq d$ we want to show that $\inf(S_1) \leq \inf(S_2)$. My attempt: Let $y_0 \in [c,d]$ be…
javacoder
  • 361
0
votes
1 answer

What does Arg {Inf I(d)} means?

I am currently studying the phase-field method for fracture modeling. In an article by Miehe -"Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field FE implementation", I came across on an identity that…
Cro Simpson2.0
  • 133
0
votes
1 answer

Is it possible to show sup$(\bigcup_{n\in\mathbb{N}}A_n)=$sup$\{$sup$A_n|n\in\mathbb{N}\}$ by induction?

Show that for every indexed family of subsets $A_n\subset\overline{\rm \mathbb{R}}$ for $n\in \mathbb{N}$ sup$(\bigcup_{n\in\mathbb{N}}A_n)=$sup$\{$sup$A_n|n\in\mathbb{N}\}$ I wanted to Show the equality by induction I assume that the equality holds…
RM777
  • 987
1 2 3
…
11 12