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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).

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Find a set A that satisfies the following

Find a set A with a order relation such that: $$\forall a, b,c \in A, \inf({\sup({a,c}),b}) = \sup({\inf({a,b}),\inf({a,c})})$$ It's easy to find a set A of two or one element that satisfies this, but ...
Carinha logo ali's user avatar
-1 votes
0 answers
23 views

Calculus with supremum infimum

Let A be a subset of $\Bbb R$.Let $f,g : A \to \Bbb R $ that bounded above and below. In a part a of an exercise I struggle to prove one of the following: $\min_{A} f-\inf_{A} g \le \sup_A {|f-g|}$ $\...
Amit Dahan's user avatar
1 vote
1 answer
127 views

Analysis of Derivatives in a Sequence of Concave Functions

Consider a sequence $\{h_n(t)\}_{n=1}^\infty$ of concave functions defined on $\mathbb{R}$. Each function $h_n$ in the sequence is differentiable at $t = 0$. We know the following about the sequence: ...
Snowball's user avatar
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Understanding infimum and jordan measure cover by open and closed set

I'm self-studying the Jordan measure and its definitions. I have a question regarding how the outer measure $ m_o(E) $ behaves when $ E $ is a closed interval and we restrict the elementary set to be ...
eugene's user avatar
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3 votes
1 answer
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Explain step in proof of $\lim_{n \to \infty} a_n = b \iff \limsup_{n \to \infty} a_n = \liminf_{n \to \infty} a_n = b$.

I'm trying to understand a step in the forward direction of the proof of the theorem $\lim_{n \to \infty} a_n = b \iff \limsup_{n \to \infty} a_n = \liminf_{n \to \infty} a_n = b$. First, to clarify, ...
TheSenate's user avatar
  • 596
2 votes
1 answer
25 views

The join of two set partitions in the refinement order

Let $X$ be a set. The set $\Pi_X$ of all partitions of $X$ is partially ordered via the refinement order, which is defined by $\alpha \leq \beta$ if and only if for each block $A\in \alpha$ there is a ...
azimut's user avatar
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1 vote
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Show that the supremum of a family of continuous functions is lower semicontinuous.

Let $(X, \mathcal{T})$ be a topological space. Consider a family $\{f_i \mid i \in I\}$ of continuous functions $f_i : X \rightarrow \overline{\mathbb{R}}.$ Define $M : X \rightarrow \overline{\mathbb{...
thecountofmontecristo's user avatar
1 vote
1 answer
41 views

The problem of equality and Hilbert space

Problem: Let $H$ be a real Hilbert space. $x_k\in H$, $a_k\in \mathbb{R} \; (k=\overline{1,n})$. Prove that $$\sup_{\sum a_k^2\leq 1}\left \| \sum_{k=1}^{n}a_kx_k \right \|=\sup_{\left \| x \right \|\...
Dmitry's user avatar
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1 vote
1 answer
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Vector Norm $\vert\vert v\vert\vert_V$ expressed as supremum of $\vert Lv\vert$ over all bounded operators L with $\vert\vert L\vert\vert_{op}\leq 1$

Let $V$ be a normed vector space with norm $\vert\vert\cdot\vert\vert_V$. How can I show that for all $v\in V$ we have $$\vert\vert v\vert\vert_V = \sup\{\vert Lv\vert \:\colon\: L\in \text{Hom}(V,\...
Apollo13's user avatar
  • 567
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0 answers
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A supremum/integral inequality

Let $I$ be an arbitrary index set and for each $i\in I$ let $f_i:\mathbb{R}^n\to [0,\infty)$ be a measurable function. For each $i\in I$ and $n\in \mathbb{N}$ let $f_i^n:\mathbb{R}^n\to [0,\infty)$ be ...
Pong's user avatar
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1 vote
0 answers
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Supremum of nineteen hundred eighty-two derivative of some function

Prove that $$\sup\left|\left(\frac{\sin x}{x}\right)^{(1982)}\right|=\frac{1}{1983}$$ It is $1982$ derivative of $\frac{\sin x}{x}$. I tried to find a general formula for n-th derivative of $\frac{\...
perenqi's user avatar
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37 views

Suppose $∅ ≠ S ⊆ \mathbb{R}$ and is bounded below by $K$. Prove that if $M = \inf~S$ and $\epsilon > 0$ then $∃x ∈ S$ ∋ $x < M + \epsilon$

My solution is as follows, $M = \inf S$ $⇒$ $M \le x$ $∀x$  $K$ lower bound ⇒ $K \le x$ $∀x$  Combining inequalities,  ($M \le x$) ∧ ($K \le x$) ⇒ ($M \le K \le x$) $∨$ $(K \le M \le x)$ $∀x$  Note ...
Chen Wu's user avatar
  • 11
2 votes
1 answer
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If the suprema of a decreasing sequence converges or goes to $+\infty$ then there is a subsequence converge to the limit of suprema or $+\infty$

I have difficulties convincing myself if the following statement is valid: Let $\{\alpha_n\}_{n=1}^{\infty}$ be a sequence. For each $n$, define $\beta_n$ by letting $\beta_n = \{\alpha_n,\alpha_{n+1}...
Beerus's user avatar
  • 1,595
3 votes
1 answer
141 views

Proving that the function $f(x) := |x|^{\frac{\lambda - n}{p}} (1- \psi(x))$ satisfies two specific properties related with limits and supremums.

Let $1 \leqslant p < \infty$ and $0 < \lambda < n$, where $n \in \mathbb N$ is an arbitrary fixed integer that stands for the dimension of the euclidian space $\mathbb R^n$. In everything ...
xyz's user avatar
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8 votes
2 answers
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What is the $\inf$ and $\sup$ of the area of quadrilateral given its sides length?

Now asked on MO here. Given the length of the sides of a quadrilateral $a,b,c,d$ the area of the quadrilateral is less than or equal to $\frac{(a+b+c+d)^2}{16}$ i.e it is an upper bound of the area ...
pie's user avatar
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1 vote
1 answer
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A partially ordered set has all suprema iff it has all infima

Let $(P, \leq)$ be a partially ordered set. We will show that every nonempty set bounded above in $P$ has a supremum iff every nonempty set bounded below in $P$ has an infimum. Obviously, it suffices ...
Smiley1000's user avatar
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6 votes
0 answers
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Questions About Four Definitions of The Upper and Lower Limits of A Sequence

Related questions have been posted here and here. Background I have seen the following four definitions of the upper and lower limits of a sequence from textbooks and MSE posts: Definition 1$\quad$ [...
Beerus's user avatar
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1 vote
1 answer
45 views

Riemann Sum with Supremum [closed]

Suppose we have $f(t, \mathbf{s}): \mathbb{R} \times \mathbb{R}^{g}$ where $\mathbf{s} \in [0, 1]^{g}$ with $g \in \mathbb{Z}_{+} \cup \infty$, $t \in \mathbf{t}$ where $\mathbf{t}$ is a set of tagged ...
SiegAndy's user avatar
1 vote
0 answers
31 views

Riemann Integration and Supremum

Suppose we have $f(t, \mathbf{s}): \mathbb{R} \times \mathbb{R}^{g}$ where $\mathbf{s} \in [0, 1]^{g}$ with $g \in \mathbb{Z}_{+} \cup \infty$, $t \in \mathbf{t}$ where $\mathbf{t}$ is a set of tagged ...
SiegAndy's user avatar
1 vote
0 answers
32 views

Upper bound for supremum of Lipschitz function

Given a Lipschitz function $L$, I want to show the following: For $\gamma >0$, let $\chi(\gamma)$ be an equidistant partition on $[0,1]$ with grid length $n^{-\gamma}$ where $n \ge 1$. Then \begin{...
WeakLearner's user avatar
  • 6,096
6 votes
4 answers
164 views

Prove that $\mu_*\leq\mu^*$ where the definition of an inner and outer measure is induced by a measure

I know that similar question has been asked and answered here, here, and here. But I am looking for a different proof based on different definitions. We have the following definition of an outer and ...
Beerus's user avatar
  • 1,595
1 vote
1 answer
29 views

Prove that if a curve y(x) is concave, (dy/dx, y-intercept of tangent to curve with slope dy/dx) are the coordinates of g(t), the supremum of y-tx

The problem description is as follows: Suppose there is a concave function $y(x)$. Now, suppose that we're interested in plotting a curve $g(t) = \sup[y - tx]$. Prove that $g(dy/dx)=y_t-tx_t$, where $...
ZED's user avatar
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1 answer
64 views

$\sup_x(y \cdot f(x)+b(x)-1/2\lVert f(x) \rVert^2)$ convex and regular with respect to $y$?

I'm studying the article "Brenier, Y. (1987) Décomposition polaire et réarrangement monotone des champs de vecteurs" (polar factorization and monotone rearrangement of vector-valued ...
Oersted's user avatar
  • 159
2 votes
0 answers
35 views

Determine the infimum of a function

I consider the function $H: \mathbb{R}\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ defined by $$ H(x,u,p) = p(-xu + \frac{1}{2}u^2) $$ I would like to find its infimum with respect to the second ...
G2MWF's user avatar
  • 1,339
1 vote
1 answer
104 views

How to prove that $ {\displaystyle \sup_{x \in \mathbb R^n, \, r > 0} r^{-\lambda} \int_{B(x,r)} |f_\alpha(y)|^p \, dy < \infty}$?

Let $1 \leqslant p < \infty$ and $0 < \lambda < n$. Consider the function $f_\alpha \colon \mathbb R^n \to \mathbb R$ defined by $$ f_\alpha(x) := \|x\|^{\frac{\lambda - n + \alpha}{p}} \chi_{...
xyz's user avatar
  • 1,069
3 votes
0 answers
65 views

How to prove that ${\displaystyle \, \, \lim_{r \to 0} \, \sup_{x \in \mathbb R^n} \, r^{-\lambda} \int_{B(x,r)} |f_\alpha(y)|^p \, dy = 0}$?

Let $1 \leqslant p < \infty$ and $0 < \lambda < n$. Consider the function $f_\alpha \colon \mathbb R^n \to \mathbb R$ defined by $$ f_\alpha(x) := \|x\|^{\frac{\lambda - n + \alpha}{p}} \chi_{...
xyz's user avatar
  • 1,069
0 votes
1 answer
24 views

An inequality involving infimums of the scaled $k$-th moment and of a scaled moment-generating function.

Let $X$ be a non-negative r.v., prove that \begin{equation} \inf_{k\in\mathbb{Z}_+} \frac{\mathbb{E}[X^k]}{t^k}\leqslant \inf_{\lambda\geqslant 0}\frac{\mathbb{E}[e^{\lambda X}]}{e^{\lambda t}},\;\...
o.spectrum's user avatar
  • 1,170
1 vote
1 answer
36 views

Assuming that $(f_n)_n$ is bounded, does $\sup_{n\in\mathbb N}||f_n||_H <+\infty$ hold?

Let $(H, \|\cdot\|)$ be a Hilbert space and let $(f_n)_n$ be a bounded sequence in $H$. I was wondering if this information is enough to conclude that $$\sup_{n\in\mathbb N}\|f_n\|_H <+\infty.$$ I'...
Physics user's user avatar
2 votes
1 answer
41 views

A supremum on orthogonal matrices

I'm working on a problem where I want to find the supremum over the orthogonal group $O_n(\mathbb{R})$ of the sum of the upper triangular elements of matrices in this group, specifically we want to ...
Jainko's user avatar
  • 31
1 vote
1 answer
48 views

$\sup$ and $\inf$ of $\{a - \frac1n|n\in \mathbb N^*\}$

I have looked at similar questions on here but still cannot reach a conclusion for this particular one. I have gathered that as n increases, $\frac1n$ tends to $0$, meaning $a - \frac1n$ tends to $a$. ...
i.diazr's user avatar
  • 49
1 vote
1 answer
86 views

Property of lower semicontinuous functions

I am trying to prove that f is sequentially lower semicontinuous at x if and only if $f(𝑥)=\sup_{r>0}\inf_{y \in B(x,r)}f(y)$. Following the proof of ($F$ is lower semicontinuous $\iff F(x)=\sup_{...
Sharon Puthuparambil's user avatar
0 votes
1 answer
97 views

Showing that the supremum of a given function is finite.

Consider arbitrary elements $n \in \mathbb N$ and $ \lambda \in \mathbb R$ such that $0 < \lambda < n$. Problem. My goal is to prove that the supremum $$ \sup_{r > 0} f(r) $$ is finite, where ...
Temirbek Alikhadzhiyev's user avatar
2 votes
1 answer
42 views

How can i show this infimum equality $A_{s-}^{-1}=\inf\{t\geq 0: A_t\geq s\}$?

Let $s\mapsto A_s$ be an increasing right continuous function, $s\geq 0$. Then define $A_s^{-1}:=\inf\{t\geq 0: A_t>s\}$ with the convention $\inf\{\emptyset\}=\infty$. By definition the map $s\...
user123234's user avatar
  • 2,915
0 votes
0 answers
20 views

Supremum of the Support of a Distribution Function

I am given the following definition of the right endpoint $x_F$ of the distribution function $F$: $x_F = sup\{x \in \mathbb R: F(x) < 1\}$ I know that the right endpoint of a discrete uniform ...
asdf1234's user avatar
0 votes
0 answers
49 views

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 ...
abelian25's user avatar
0 votes
2 answers
81 views

$sin(1/x)$: expectation vs infimum

I want to show that $$ \underset{Q \in \mathcal{P}^0}{\inf} \ \{\mathbb{E}_{x \sim Q} \sin(1/x)\} > \inf_{x \in \mathbb{R}}\{ \sin(1/x)\} = -1$$ holds where $\mathcal{P}^0$ is the set of all ...
independentvariable's user avatar
0 votes
2 answers
70 views

If $f:\mathbb R\to\mathbb R$ is such that $f>0$ a.e., do $\inf_{\mathbb R} f >0$ and $\sup_{\mathbb R} f>0$?

Let $f:\mathbb R\to\mathbb R$ be a continuous function such that $f>0$ for almost every $x\in\mathbb R$. Does one can deduce from this that both $\inf_{x\in\mathbb R} f(x) >0$ and $\sup_{x\in\...
Physics user's user avatar
1 vote
1 answer
79 views

Application of weak maximum principle.

Fix any open, bounded set $U \subset \mathbb{R}^n$ and suppose that $u \in C^2(U) \cap C(\bar{U})$ is a solution of $$-\Delta u = f \,\,\,\text{in}\,U$$ $$u=g \,\,\,\,\,\,\text{on}\,\,\,\partial U,$$ ...
Lilili123's user avatar
  • 139
2 votes
1 answer
48 views

Are these two questions related?

$f$ and $g$ are bounded functions with common domain$D$ $\sup\limits_{x\in D}\big\{f(x)+g(x)\big\}\leqslant\sup\limits_{x\in D}f(x)+\sup\limits_{x\in D}g(x).$ Let both $S $and $T$ be non-empty subsets ...
Linhao231035's user avatar
0 votes
0 answers
33 views

Easy direction for exchanging limit and supremum

Let $X$ be a metric space and $(f_n)$ be a sequence of continuous functions with $f_n: X \rightarrow \mathbb{R}$. I know that, in general, $\sup_{x \in X}\lim_{n\in \mathbb{N}}f_n(x)\ne \lim_{n\in \...
12345's user avatar
  • 187
1 vote
1 answer
90 views

inequality for supremum [closed]

Consider two bounded sequences $\{|A_i|\}_{i = 1}^{\infty}$ and $\{|B_i|\}_{i = 1}^{\infty}$, for some $\epsilon > 0$, suppose $\sup_{i \in \mathbb{N}}||A_i| - |B_i|| < \epsilon$ where $\mathbb{...
Marshall's user avatar
3 votes
1 answer
311 views

Some doubtful implication for mathematical analysis.

Let, $f(x),g(x),f_1(x),g_1(x)$ are positive real valued bounded and continuous functions on domain of non-negative reals and also having range between $0$ and $1$. And, also, $f_1(x),g_1(x)$ are ...
A learner's user avatar
  • 2,841
0 votes
1 answer
61 views

Find supremum and infimum of $f(x)$ such that $\lim_{x\to\pm\infty}\frac{f(x)}{|x|}=\infty$

If $\lim_{x\to\pm\infty}\frac{f(x)}{|x|}=\infty$, with $f:\mathbb{R}\to\mathbb{R}$ differentiable, then can I say by applying de l'Hopital that $\lim_{x\to\infty}f'(x)=+\infty$ and $\lim_{x\to-\infty}...
axi's user avatar
  • 245
3 votes
4 answers
398 views

Supremum of sum of functions strictly less than the sum of the functions' suprema (sup f+g < sup f + sup g) [duplicate]

Given the bounded functions $f, g: X \rightarrow \mathbb{R}$, I understant that $\sup (f+g) \leq \sup f + \sup g$. However I can't think of an example where $\sup (f+g) < \sup f + \sup g$, strictly....
Eduarda's user avatar
  • 41
0 votes
0 answers
41 views

$2.47$ Rudin non-connectedness in $\mathbb R$ - how do we formally prove that $z_1 \notin A$ ? Consider the cases I drew in these schemes

This is theorem $2.47$ from Principles of Mathematical Analysis by W. Rudin (it's aiming to prove that in $\mathbb R$ a part $X$ is disconnected iff it doesn't verify $\forall a,b \in X$ such as $a<...
niobium's user avatar
  • 1,221
2 votes
1 answer
98 views

Essentialy bounded functions with compact support are locally integrable and satisfy aditional condition?

Consider arbitrary elements $1 \leqslant p < \infty$ and $0 < \lambda \leqslant n$. Furthermore, during this post I considered the usual Lebesgue measure over $\mathbb R^n$ and I denote the ...
Temirbek Alikhadzhiyev's user avatar
0 votes
0 answers
14 views

Recover an essential infimum given a family of measures from a process,

Suposse that for some family of probability measures $\mathcal{P}$ the process $ess\inf_{\mathbb{P}\in\mathcal{P}}\mathbb{E}^\mathbb{P}[B|\mathcal{F_t}]$ is a submartingale, for some possitive r.v. $B$...
Don P.'s user avatar
  • 314
0 votes
1 answer
35 views

If $A,B⊆X⊆ \mathbb R$, $A\cap B=∅$ $A,B$ open then $\sup A<\inf B$ or $\sup B< \inf A$

Let $X\subseteq \mathbb R$ be a metric space. I'm trying to prove that if a subset $I$ is non connected, it implies that $\exists a,b \in I$, $a<b$, $\exists x, a<x<b$ such that $x\notin I$. ...
niobium's user avatar
  • 1,221
1 vote
2 answers
88 views

Why isn't $\sup(f(x)+g(x)) = \sup f(x) + \sup g(x)$?

The claim that if $X$ is a non empty set and $f,g:X \to \mathbb{R}$ bounded functions then $$\sup_{x \in X} (f(x)+g(x)) = \sup_{x \in X} f(x) + \sup_{x \in X} g(x)$$ Is not true in general. But why is ...
MC2's user avatar
  • 751
0 votes
1 answer
63 views

Supremum notation

I hope I will be forgiven for having such a basic question. I am trying to understand Definition 1.21 of Meng. I paste the definition here: I assume this can be written with more traditional notation ...
Richard Southwell's user avatar

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