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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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Question regarding supremum of partially ordered sets

I have encountered a question while i was practicing the topic 'upper and lower bound of partially ordered sets'. Let $\mathbb{Q}$ be the set of rational numbers. Let $$ B = \{ x \in \mathbb{...
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5answers
42 views

Finding the supremum of $\left\{n^{\frac{1}{n}}\;\middle\vert\;n\in\mathbb{N}\right\}$

$\left\{n^{\frac{1}{n}}\;\middle\vert\;n\in\mathbb{N}\right\}$ What is the Supremum of the above set? I consider the function $f(x)= x^{\frac{1}{x}}$, and show that $f(x)$ is maximum when $x=e$. ...
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1answer
42 views

Showing that $\alpha\geq \beta$

Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on a complex Hilbert space $F$. Let $A,B\in \mathcal{B}(F)$. Consider the following numbers: $$\alpha=\sup_{\substack{a,b\in \mathbb{C}...
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1answer
17 views

Supremum equals maximum on a subset of natural numbers

I'm wondering if $\sup_{x \in M} f(x) = \max_{x \in M} f(x)$ holds when $f$ is some arbitrary function and $M = \{0,1,\dots,n\}$ for some $n \in \mathbb N$. My idea is that $M$ is closed and bounded ...
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1answer
14 views

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)?
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1answer
14 views

Double supremum of double sequence

This is from "Measures, Integrals and Martingales" by R.L. Schilling, page 28, exercise 4.6ii. Prove that for any double sequence $\beta_{ij}, i, j \in \mathbb{N}$ of real numbers, we have $$ \sup_{...
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1answer
40 views

Finding the supremum and infimum of $\left\{n^{\frac{1}{n}}\;\middle\vert\;n\in\mathbb{N}\right\}$

$\left\{n^{\frac{1}{n}}\;\middle\vert\;n\in\mathbb{N}\right\}$ What is the supremum and infimum of the above set? The set is $\left\{1, 2^{\frac{1}{2}}, 3^{\frac{1}{3}},....\right\}$ Now, $n^{\frac{...
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2answers
30 views

monotonic function can only have simple discontinuity

I am self-studying Rudin, Principles of Mathematical Analysis. I am having trouble going through the theorem saying that monotonical functions can only have simple discontinuity, i.e., Suppose $f$ is ...
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2answers
31 views

Is $\inf A $ and $\sup A $ belong $\bar A$?

Let $A$ be a anonempty and bounded subset of $\mathbb{R}$. Now take $A= (0,1)$ in the discrete topology of $\mathbb{R} $. My question is that : Is $\inf A $ and $\sup A $ belong $\bar A$ ? ...
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1answer
42 views

Supremum of uniformly converges functions goes to the supremum of the limiting function

Let ${ f_n\left(x\right):[0,1] \to \mathbb{R} }$ uniformly convergent to function $f(x)$ which is bounded on the interval $[0,1]$. Prove the following: ${ \lim_{n\to\infty} sup_{[0,1]} f_n\left(x\...
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1answer
41 views

If $f:[0,\infty)\to\mathbb R$ is continuous and $\tau=\inf\left\{t>0:f(t)>\varepsilon\right\}$, then $f(\tau\wedge t)\le\varepsilon$ for all $t\ge0$

Let $f:[0,\infty)\to\mathbb R$ be continuous, $\varepsilon>0$ and $$\tau:=\inf\left\{t>0:f(t)>\varepsilon\right\}.$$ Are we able to show that $f(\tau\wedge t)\le\varepsilon$ for all $t\ge0$ (...
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1answer
16 views

For $a,b \in \mathbb{R}$ fixed supremum and infimum property

For $a,b \in \mathbb{R}$ fixed and $S$ is a bounded set above and below I want to prove the following: If $a \geq 0$ then $\inf aS+b = a \inf S +b$ and $\sup aS+b = a \sup S +b$ I am not having ...
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1answer
121 views

Is supremum / infimum concept an axiom. ( Equivalence between infimum's definitions ) [closed]

Earlier title was : 'Equivalence between infimum's definitions'. But, that changed as failure occurred in proof as shown below. I am sorry, if naive, but this is my experience with proving the part (...
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0answers
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Is there a way to find the largest $n\in \Bbb N$ s.t. f is n times differentiable?

Is there a method to find sup$\{n\in \Bbb N: \text{f is n times differentiable} \}$?
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2answers
25 views

Uniform convergence of a sequence of functions 4

Prove that the sequence $\left((nx)/(1+4n^2x^2)\right)_{n\in\mathbb N}$ is not uniformly convergent on $(-a,a)$, where $a > 0$ My attempt: $\lim_{n\to\infty}(nx)/(1+4n^2x^2) = 0 = f(x)$ Now, ...
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1answer
73 views

Questions based on $\epsilon$ based definition of Supremum.

Need help in vetting my answers for questions in sec 2.5 in chap. 2 (page 8) in CRM series book by MAA: Exploratory Examples for Real Analysis, By Joanne E. Snow, Kirk E. Weller. The question ...
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1answer
61 views

Proving equivalence between $\epsilon$ based & $lub$ definition of supremum.

Based on $\epsilon$ have a new definition of supremum: Let there be a nonempty set $X$ with supremum $s$, then $X\cap(s - \epsilon, s]\ne \emptyset, \,\, \forall \epsilon\gt 0$. The conventional ...
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1answer
68 views

Supremum proof based on $\epsilon$.

Need help in vetting my answers for questions in section 2.3, 2,4 (on page # 7,8) in chap. 2 (page 7) in CRM series book by MAA: Exploratory Examples for Real Analysis, By Joanne E. Snow, Kirk E. ...
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3answers
49 views

Supremum, infimum of $|A|$.

Let $A$ be a nonempty subset of the real numbers. Define the set $|A|$ to be $|A|:= \{|x| : x \in A\}$. If the set $A$ is bounded, is the set $|A|$ bounded? If not, give an example. If so, by what? ...
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1answer
75 views

Find for given upper-bound, epsilon.

Need help in vetting my answers for Q. 3,4,5 in section 2.2.2 in chap. 2 (page 7) in CRM series book by MAA: Exploratory Examples for Real Analysis, By Joanne E. Snow, Kirk E. Weller. Also, this post ...
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1answer
22 views

Limit of sequence of infimums of a continuous function over converging sequence of sets equals infimum of the function over the limit of the set.

Let $X$ be any topological space. Let $C_n$ be monotone decreasing sequence of compact sets in $X$ that converges to a non-empty compact set $C$. That is $\bigcap_{n\geq 0}C_n = C$ and $C_{n+1}\...
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1answer
83 views

Find for given supremum, epsilon.

Need help in vetting my answers for Q. 1,2 in section 2.2.2 in chap. 2 (page 7) in CRM series book by MAA: Exploratory Examples for Real Analysis, By Joanne E. Snow, Kirk E. Weller. . Q. 1: For ...
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2answers
151 views

Doubts about supremum.

Need help in vetting my answers for Q. 1 in chap. 2 in CRM series book by MAA: Exploratory Examples for Real Analysis, By Joanne E. Snow, Kirk E. Weller. . Let $S_1 = \frac n{n+1} : n \in \mathbb{...
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2answers
51 views

If a < b, what is the supremum of (a,b)

Let $𝑎 \in \mathbb{R} $ and $𝑎<𝑏$. Make a conjecture about the supremum of (a, b). My conjecture was this: Since $𝑎<𝑏$ then $𝑠𝑢𝑝(𝑎,𝑏)=𝑏$. Proof: Let $c = sup(a,b)$. Let $d$ be the ...
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1answer
20 views

Finding set of upper bounds

In strict lexicographical ordering :Given the following Find, if possible, three explicit upper bounds for $C = \{(x, y) \in \mathbb{R}^2 | y < 0\}$. and write the set of upper bounds for $C$ in ...
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1answer
60 views

Supremum, bounds of subsets of real numbers.

Need help in vetting my answers for the questions in CRM series book by MAA: Exploratory Examples for Real Analysis, By Joanne E. Snow, Kirk E. Weller. Suppose there are two nonempty subsets of ...
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1answer
18 views

Is there a common way to check essential supremum?

Is there a common way to check essential supremum? Particularly, my problem is in figuring out how to demonstrate that the measure is zero.
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1answer
63 views

Find for given set $A$, $|A|$ set's supremum & bounds.

Need help in vetting my answers for the questions in CRM series book by MAA: Exploratory Examples for Real Analysis, By Joanne E. Snow, Kirk E. Weller. Let $A$ be a nonempty subset of the ...
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1answer
31 views

A question about infimum: $k\leq a_i\implies k\leq\inf_i a_i$?

Let $k$ be a constant real number, let $A\subseteq \mathbb{R}$ be a set with countable cardinality. if $k\leqslant a$ for all $a\in A$, is it true that $k\leqslant \inf A$? Must we have $k<a$ for ...
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1answer
156 views

Book questions on supremum, bounds.

Need help in vetting my answers for the questions in CRM series book by MAA: Exploratory Examples for Real Analysis, By Joanne E. Snow, Kirk E. Weller. I am particularly suspicious about my results ...
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0answers
20 views

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\} \...
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3answers
39 views

Infimum and supremum for $\{ x \in \Bbb R\mid x^2-2x-1 < 0 \} $

I've been asked to obtain the infimum, supremum, minimum and maximum for: $\{ x \in \Bbb R \mid x^2-2x-1 < 0 \} $ and $\{x^2-2x-1\mid x \in \Bbb R\} $ So for the first, I used the quadratic ...
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1answer
59 views

Supremum of product of sets $A,B$.

Taken from sec. 1.4.1 of the book by Mary Hart, titled: Guide to Analysis. Let $A, B$ be two non-empty sets of real numbers with supremums $\alpha, \beta$ respectively, and let the sets $A + B$ and ...
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0answers
90 views

Show that $\alpha+ \beta$ is the supremum of $A + B$. [duplicate]

Taken from sec. 1.4.1 of the book by Mary Hart, titled: Guide to Analysis. Let $A, B$ be two non-empty sets of real numbers with supremums $\alpha, \beta$ respectively, and let the sets $A + B$ and ...
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1answer
27 views

Continuity and Lipschitz Property of Infimal Convolution

If $f$ is convex and Lipschitz-continuous on a real Hilbert space $H$ and $g$ is lsc and convex then is the infimal convolution $f\square g$ Lipschitz?
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1answer
42 views

How to prove this simple property for two sets?

We are given two vectors, $a = (a_1,\dots,a_n)$ and $b = (b_1,\dots,b_n)$ such that $0 \le a_i=b_i \le \varDelta$ for $i=1,\dots,n$. We want to modify each of these vectors in an iterative procedure ...
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2answers
62 views

An intriguing supremum… which may be inaccurate

I found the following exercise and I would like to know whether the property is true or not, and above all how to prove it: for $z$ so that $\Im z>0$, $$\sup_{t\in\mathbb{R}}\left|\frac{t-i}{t-z}...
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0answers
48 views

Rudin Theorem 1.11 “counterexample” (where is my mistake?)

This is with reference to Principles of Mathematical Analysis, Third Edition by Walter Rudin. The theorem establishes that every ordered set with the least-upper-bound property also has the greatest-...
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4answers
118 views

Find $\limsup _{n\to\infty} \bigl(\frac{2\cdot5\cdot8\cdot\cdots\cdot(3n-4)}{3^nn!}\bigr)^{1/n}$

Find $\limsup_{n\to\infty}(\frac{2\cdot5\cdot8\cdot\cdots\cdot(3n-4)}{3^nn!})^{1/n}$ I've tried multiplying the nominator and the denominator by what is lacking for there to be $3n!$ in the nominator,...
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1answer
24 views

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(...
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3answers
53 views

If $\inf(A) < \sup (B) $ then there is some $a \in A$ and $b \in B$ such that $a<b$

Proposition: Let A be a subset of R which is bounded below. Let B be a subset of R which is bounded above. If $\inf(A) < \sup (B) $ then there is some $a \in A$ and $b \in B$ such that $a<b$. ...
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0answers
17 views

Inequality using pseudometric

I have the following question. Let $I$ be a finite set, $\{S_i\}_{i \in I}$ be a family of sets and $S = \prod_{i \in I} S_i$ the Carthesian product of this family of sets. Consider the bounded ...
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3answers
28 views

decide if 3 variables function has a minimum and maximum value

There is a question in my textbook that says decide if the function $$f(x,y,z) = 2x^3+2y^3+2z^3-3xy-3yz-3zx$$ has a minimum and maximum value on R^3 And it says that the solution is to look at ...
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4answers
33 views

infimum of two multiplied sequences the same as the product of the infimum of one sequence and the infimum of another

What is a counter example for $\inf(x_n\cdot y_n) = \inf(x_n)\cdot \inf(y_n)$ if $x_n$ and $y_n$ are bounded below.
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1answer
54 views

Double Supremum Conflicting Conventions

We read $$\sup_{x\in X}(\sup_{y\in Y} f(x,y))$$ left to right (there's really no other choice), but Question 1: doesn't this contradict the following conventions? Stuff inside parentheses is ...
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1answer
20 views

Uniform convergence of $ \frac{n\cdot \sin \left(x\right)}{1+n\cdot \cos \left(x\right)} $

Check uniform convergence of $$f_n(x) =\frac{n\cdot \sin \left(x\right)}{1+n\cdot \cos \left(x\right)} $$ on $[-a;a$] where $0<a< \frac{\pi}{2}$ ok so firstly I check pointwise convergence $$ \...
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1answer
24 views

$sup_{x\in R} |\sum λ_n e^{-|x-n|}|\leq \frac{e+1}{e-1}$

Let $(λ_n)$ be a sequence convergent to zero. If $|λ_{n}|\leq 1$ for each $n\in \mathbb{N} $ then $sup_{x\in \mathbb{R}} |\sum λ_n e^{-|x-n|}|\leq \frac{e+1}{e-1}$ Could you give me a suggestion to ...
0
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0answers
20 views

Moving Supremum inside conditional expectation

Let $\{Z_l\}_{l \in \Lambda},X$ be square-integrable random-variables, and $f:\mathbb{R}^2\rightarrow \mathbb{R}$ be a continuous function, such that $ \sup_{l \in \Lambda}\mathbb{E}[f(Z_l,X)]<\...
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2answers
35 views

Let $\emptyset ≠A⊆F$. $\exists \max(A) \implies \exists \sup(A)$ and $\sup(A)=\max(A)$. [closed]

Let $∅≠A⊆F$. Prove that if $\max(A)$ exists, then $\sup(A)$ exists and $\sup(A)=\max(A)$. How do I prove such statement?
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3answers
64 views

Find a set of rational numbers where $\sqrt{3}$ is the infimum. Prove it

I thought about the expansion of $\sqrt{3}$ as a series. But I didn't get anything useful. Also, I thought about a set with irrational numbers instead of rational numbers. What is the general idea ...