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).
0
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1answer
39 views
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
$$
B = \{ x \in \mathbb{...
5
votes
5answers
42 views
Finding the supremum of $\left\{n^{\frac{1}{n}}\;\middle\vert\;n\in\mathbb{N}\right\}$
$\left\{n^{\frac{1}{n}}\;\middle\vert\;n\in\mathbb{N}\right\}$
What is the Supremum of the above set?
I consider the function $f(x)= x^{\frac{1}{x}}$, and show that $f(x)$ is maximum when $x=e$.
...
2
votes
1answer
42 views
Showing that $\alpha\geq \beta$
Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on a complex Hilbert space $F$.
Let $A,B\in \mathcal{B}(F)$. Consider the following numbers:
$$\alpha=\sup_{\substack{a,b\in \mathbb{C}...
0
votes
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 ...
0
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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)?
0
votes
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
$$ \sup_{...
0
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1answer
40 views
Finding the supremum and infimum of $\left\{n^{\frac{1}{n}}\;\middle\vert\;n\in\mathbb{N}\right\}$
$\left\{n^{\frac{1}{n}}\;\middle\vert\;n\in\mathbb{N}\right\}$
What is the supremum and infimum of the above set?
The set is
$\left\{1, 2^{\frac{1}{2}}, 3^{\frac{1}{3}},....\right\}$
Now, $n^{\frac{...
1
vote
2answers
30 views
monotonic function can only have simple discontinuity
I am self-studying Rudin, Principles of Mathematical Analysis. I am having trouble going through the theorem saying that monotonical functions can only have simple discontinuity, i.e., Suppose $f$ is ...
0
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2answers
31 views
Is $\inf A $ and $\sup A $ belong $\bar A$?
Let $A$ be a anonempty and bounded subset of $\mathbb{R}$.
Now take $A= (0,1)$ in the discrete topology of $\mathbb{R} $.
My question is that :
Is $\inf A $ and $\sup A $ belong $\bar A$ ?
...
1
vote
1answer
42 views
Supremum of uniformly converges functions goes to the supremum of the limiting function
Let ${ f_n\left(x\right):[0,1] \to \mathbb{R} }$ uniformly convergent to function $f(x)$ which is bounded on the interval $[0,1]$. Prove the following:
${ \lim_{n\to\infty} sup_{[0,1]} f_n\left(x\...
0
votes
1answer
41 views
If $f:[0,\infty)\to\mathbb R$ is continuous and $\tau=\inf\left\{t>0:f(t)>\varepsilon\right\}$, then $f(\tau\wedge t)\le\varepsilon$ for all $t\ge0$
Let $f:[0,\infty)\to\mathbb R$ be continuous, $\varepsilon>0$ and $$\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$ (...
0
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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 ...
0
votes
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 (...
0
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0answers
36 views
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} \}$?
0
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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, ...
0
votes
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 ...
0
votes
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 ...
0
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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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votes
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? ...
0
votes
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 ...
1
vote
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}\...
0
votes
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 ...
4
votes
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{...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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.
2
votes
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 ...
-2
votes
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 ...
3
votes
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 ...
0
votes
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\}
\...
0
votes
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 ...
1
vote
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 ...
1
vote
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 ...
2
votes
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?
2
votes
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 ...
1
vote
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 $\Im z>0$,
$$\sup_{t\in\mathbb{R}}\left|\frac{t-i}{t-z}...
2
votes
0answers
48 views
Rudin Theorem 1.11 “counterexample” (where is my mistake?)
This is with reference to Principles of Mathematical Analysis, Third Edition by Walter Rudin. The theorem establishes that every ordered set with the least-upper-bound property also has the greatest-...
4
votes
4answers
118 views
Find $\limsup _{n\to\infty} \bigl(\frac{2\cdot5\cdot8\cdot\cdots\cdot(3n-4)}{3^nn!}\bigr)^{1/n}$
Find $\limsup_{n\to\infty}(\frac{2\cdot5\cdot8\cdot\cdots\cdot(3n-4)}{3^nn!})^{1/n}$
I've tried multiplying the nominator and the denominator by what is lacking for there to be $3n!$ in the nominator,...
0
votes
1answer
24 views
mathematical conversion
I want to understand, how i get from the left to the right side in the following inequaltiy:
$$\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(...
1
vote
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 ...
0
votes
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 ...
0
votes
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.
0
votes
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 ...
1
vote
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
$$ \...
0
votes
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
votes
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)]<\...
1
vote
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?
0
votes
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 ...
