Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit 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).

1
vote
1answer
15 views

Show that support function of a set $S$ and support function of the closure of that set $\bar{S}$ are equal.

Let $S\subseteq \mathbb{R}^n$. The support function of set $S$ is defined as the following $$ \sigma_S(x)=\sup_{y \in S} x^Ty $$ where $x \in \mathbb{R}^n$. Show that $\sigma_S(x)=\sigma_{\bar{S}}(...
0
votes
1answer
32 views

Find infA when $A=\{\frac{m^2-n}{m^2+n^2}:m,n\in \Bbb N,m>2n>1\} $

$A=\left\{\frac{m^2-n}{m^2+n^2}:m,n\in \Bbb N,m>2n>1\right\} $ $\sup A=1$ $1$ is upper bound: $\frac{m^2-n}{m^2+n^2}\le1\\m^2-n\le m^2+n^2 \\ -n \le n^2 \mbox{ true}$ And for $n=1$,$ \lim \...
1
vote
1answer
14 views

Is the supremum of an almost surely continuous random function random variable?

Let {$X_t, t\in[0,1]$} on {$R, \mathfrak B(R) $} be random, almost surely continuous, function. How to show that $X^+=sup_{t \in[0,1]} X_t$ is random variable ? Perhaps here I can say that $X_t$ it ...
0
votes
1answer
51 views

sup and inf of $A=\left\{\frac{m^4+2n^2}{2m^2-m^2n+n^2}:m,n \in\Bbb N\right\}$

$$A=\left\{\frac{m^4+2n^2}{2m^2-m^2n+n^2}:m,n \in\Bbb N\right\}$$ $\sup A$ doesn't exist, because $A$ is not bounded above: $$\begin{align}n&=1 \\ \frac{m^4+2}{m^2+1} &\le M \\ \frac{m^4+2}{...
0
votes
1answer
14 views

Prove property of $A$ implies $\inf A \le 2$

Set $A$ is a proper subset of reals and has a property that for every $a \in A$ there exists $b \in A$ such that $a \ge 2b - 2$. How to prove that $\inf A \le 2$?
1
vote
1answer
60 views

Real analysis use of sup and inf

Let $A$ be a nonempty set of positive real numbers, and consider the set $$B = \{ \tfrac1a \mid a\in A \}.$$ I have to prove the following: 1) The set $B$ is bounded above if and only if $\inf(A)&...
0
votes
1answer
8 views

norm of element in equivalent class in quotient space

If we have a quotient space $E\backslash L_0$ where $E$ is a linear normed space and $L_0$ it's subspace the norm of an element $L$ in $E\backslash L_0$ is defined as $$\lVert L\rVert = \inf_{x \in L}{...
0
votes
2answers
32 views

Is $f$ continuous at $0$?

$f : [0, 1] → \mathbb{R}$ $f(x) := \inf\{|nx − 1| : n ∈ \mathbb{N}\}$ I found that $f(x)$ is continuous on $\big(\dfrac{1}{m+1}, \dfrac1m\big]$ and that $\displaystyle \lim_{x\to 0}f(x)=0$ How ...
2
votes
1answer
35 views

Supremum and infimum of a set

I wanted to check my reasoning for the following question: Determine minimum, maximum, supremum and infimum of the set: $$B=\left\{ -\frac{1}{n} \in \mathbb{Q}: n \in \mathbb{N}_+ \right\}$$ ...
0
votes
1answer
17 views

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 ...
0
votes
0answers
12 views

$f:[0,1]\to\Bbb{R}$ has property- $\forall y\in\Bbb{R}$, either $\nexists x\in[0,1]$ s.t. $f(x)=y$ or $\exists$ exactly two such points in $[0,1]$.

My whole question looks like- A real valued function $f:[0,1]\to\Bbb{R}$ has the property that $\forall y\in\Bbb{R}$, either $\nexists x\in[0,1]$ s.t. $f(x)=y$ or $\exists$ exactly two such ...
0
votes
0answers
18 views

If maximum of an ordered, set exists, this is equal to the supremum.

Suppose we have an ordered set $A \subset M$, which has a maximum, denote it by $\max(A)$, then this maximum is equal to the supremum (the smallest of upper bounds) of $A$. I will focus on explaining ...
1
vote
2answers
38 views

what will be the supremum of $a_n $ ?

For $n \ge 1 $ let $ a_n=(1-\frac{1}{n})\sin (\frac{n\pi}{3})$, what will be the supremum of $a_n $ ? My attempt : i gots $\frac{\sqrt 3}{2}$ because $\sin\pi/ 3=\frac{\sqrt 3}{2}$ ...
1
vote
0answers
22 views

Let $f$ be a mapping, $\beta$ be an ordinal, $X=\{\alpha\mid f(\alpha)\le \beta\}$, and $\gamma=\sup X$. Is $\gamma\in X$?

Let $f:\operatorname{Ord}\to\operatorname{Ord}$ be a mapping consisting of only addition, multiplication, and exponentiation operations, and $\beta$ be an ordinal. Let $X=\{\alpha\mid f(\alpha)\le \...
2
votes
0answers
17 views

Limits, supremum and Lebesgue measure

I would like to prove this: $sup_{R\in R(x)} \frac{1}{|R|}\frac{|R \cap \epsilon B|}{|\epsilon B|}$ $\to \frac{1}{|x_1||x_2|}$ , as $\epsilon \to 0^{+}$, $\forall x_1,x_2 \neq 0$ where $R(x)$ is the ...
0
votes
1answer
32 views

Why is $f(x)=\inf \{ r \in \mathbb{Q} ;x \in U(r)\}$?

Why is $f(x)=\inf \{ r \in \mathbb{Q} ;x \in U(r)\}$ where $f$ is a real valued function from a metric space $(S,\rho)$ to $\mathbb{R}$. It is given that $$U(r)=\{x \in S;f(x) \le r \}$$ I could show ...
6
votes
1answer
41 views

What are the conditions on $f$ for $\sup f$ and $f(\sup)$ be interchangeable?

Let $f:\mathbb{R}^n \to \mathbb{R}$. Say that $f$ is magic if for every nonempty set $X$ and every functions $g_1,\dots,g_n : X \to \mathbb{R}$ the following holds: $$\sup_{(x_1,\dots,x_n) \in X^n} f(...
2
votes
1answer
52 views

The reasoning behind $\sup\{\alpha\beta+\alpha\zeta\mid\zeta<\gamma\}=\alpha\beta+\sup\{\alpha\zeta\mid\zeta<\gamma\}$

Let $\alpha,\beta,\gamma$ be ordinals. Then $\alpha(\beta+\gamma)=\alpha\beta+\alpha\gamma$. I'm been looking for some proofs of this theorem on MSE and found that the gist is the equality $\sup\{\...
1
vote
1answer
25 views

proof verification for $A_n\uparrow A \implies A-A_n\downarrow \emptyset$

For the sequence of sets $\{A_n\}_{n=1}^{\infty}$I have that $A_n\uparrow A$ i.e. $A_1\subseteq A_2 \subseteq \cdots $ and $\bigcup\limits_{i=1}^{\infty} A_{i}$ = A. I want to prove that for $B_n=A-...
0
votes
2answers
24 views

supA and infA, if $A:=\{\frac {a+b+c}{abc}: a,b,c \in \Bbb N\}$ [closed]

Find supA, and infA, if $A:=\{\frac {a+b+c}{abc}: a,b,c \in \Bbb N\}$ I think $supA=3$ and $infA=0$ Am I right?
1
vote
2answers
44 views

Supremum of $\frac{(x+y)^{2}}{2^{xy}}$

How to find the supremum of $$ \frac{(x+y)^2}{2^{xy}} $$ for $x\ge 1$ and $y\ge 1$ I have no idea how to start with it and when try to find it in WolframAlpha it only gives my numeric value.
2
votes
2answers
39 views

General topology / real analysis - Suppose $S$ is a bounded and closed nonempty subset of real numbers. Prove sup S is in S

Q. Suppose S is a bounded and closed nonempty subset of real numbers. Prove $\sup S$ is in $S$. Since $S$ is bounded and by the least upper bound property of $\mathbb R$, there exists $\sup S \in \...
0
votes
1answer
51 views

Proof of $sup\bigcup_{i\in I} A_{i}=sup\{supA_{i}, i\in I\} $

Let $I\ne\emptyset $, and $\forall_{i\in I}\ \emptyset\ne A_{i}\subseteq \Bbb R$, and $\forall_{i\in I}\exists\ c\in\Bbb R$ which is upper bound of $A_{i}$. Prove that $sup\bigcup_{i\in I} A_{i}=sup\{...
0
votes
2answers
43 views

When are the suprema of two sets equal?

This is a follow-up to my question here. Let $A$ and $B$ be sets of real numbers. My question is, under what circumstances is the supremum of $A$ equal to the supremum of $B$? Now $x$ is the ...
1
vote
1answer
27 views

When is the supremum of one set equal to the infimum of another set?

Let $A$ and $B$ be sets of real numbers. My question is, under what circumstances is the supremum of $A$ equal to the infimum of $B$? Now $x$ is the supremum of $A$ if and only if for any $\epsilon&...
1
vote
1answer
40 views

What is exactly meant with $\int f d\mu=\sup_{\rho \in \Phi^{+}_{f}}\int\rho d\mu$? Why supremum?

I have the following statements in what I'm reading: Let $(X,\mathcal{F},\mu)$ be a measure space, and let $$\mathcal{M}(X,\mathcal{F})^{+}=\{f:X\to\mathbb{R}:\text{f is }\mathcal{F}\text{-...
0
votes
1answer
44 views

Validate the proof that $x_n = {\ln n \over n}$ is bounded by $\ln2$ and find $\sup\{x_n\}$

Let $n\in \mathbb N$ and: $$ x_n = \left\{ \ln n \over n \right\} $$ Prove $x_n$ is bounded by $\ln 2$. Find $\sup\{x_n\}$ Consider nominator and denominator, $\ln n$ is growing slower than $n$ ...
0
votes
1answer
25 views

How to interpret this notation

Definition 9.1. Let $f$ and $g$ be two maps. The $C^0$-distance between $f$ and $g$, written $d_0(f,g)$, is given by $$d_0(f,g)=\sup_{x\in\mathbb{R}}\vert f(x)-g(x)\vert$$ The $C^r$-distance $d_r(f,g)$...
0
votes
4answers
40 views

Infimum and supremum of $x^5y^2z$

$A=\{x^5y^2z : x,y,z>0$ and $x+y+z=7\}$ In my mind $\sup A=5^5$ and $\inf A=0$, but how can I prove this?
0
votes
1answer
22 views

Showing that $ O \cap M = \emptyset $ or $ O \cap M = \{b\} $ Where $M$ is a set and $O$ is the set of Upper Bounds.

This is a question in the topic of Supremum/Maximum/Infimum/Minimum. Showing that $ O \cap M = \emptyset $ or $ O \cap M = \{b\} $ is true. Where $M$ is a set and $O$ is the set of Upper Bounds. ...
0
votes
1answer
21 views

Prove $x_n = \sum_{k=1}^n \frac{1}{(a+(k-1)\cdot d)\cdot(a+k\cdot d)}$ is a bounded sequence.

Let $n \in \mathbb N$ and: $$ x_n = \sum_{k=1}^n \frac{1}{(a+(k-1)\cdot d)\cdot(a+k\cdot d)} $$ Prove $\{x_n\}$ is a bounded sequence. I'm having hard time finishing the proof. Below is what i've ...
1
vote
1answer
37 views

Supremum and infimum of $\{x, y \ge 1 : \frac{xy}{3x + 2y + 1}\}$

How to find supremum and infimum of $\{x, y \ge 1 : \frac{xy}{3x + 2y + 1}\}$? I suspect that $\frac{1}{6}$ is infimum and supremum does not exist, but I dont know how to prove it using only the ...
0
votes
1answer
33 views

Does the supremum of the set $S=\{x\in\mathbb{Q}: x>2\}\subset \mathbb{R}$ exist?

Does the supremum of the set $S=\{x\in\mathbb{Q}: x>2\}\subset \mathbb{R}$ exist? If not, prove it. I don't think the supremum exist, but I don't know how to prove it.
1
vote
1answer
37 views

Please help me to understand the solution of this problem

Could anybody help me to understand the solution? $\textbf{Problem:}$Proof that $\sup(A) = \sqrt{2}$,$~~$where $A = \lbrace x \in \mathbb{Q}: x > 0, x^2 < 2 \rbrace$ For proving the right-hand ...
1
vote
1answer
34 views

Continuity of supremum of polynomial

Prove: Let $A \subseteq \mathbb{R}$ be a compact set. Prove that the function $f \colon\mathbb{R^{n+1}} \to \mathbb{R}$ $\qquad f(x_0,..., x_n) = \sup_{x\in A} \prod_{j=0}^{n} (x-x_j)$ is continuous....
1
vote
1answer
21 views

Prove the limit superior of a bounded sequence converges

Let $(a_n)_{n=1}^\infty$ be a bounded sequence and $b_n = \sup\{a_k\ |\ k \geq n\}$. Prove $b_n$ converges. This is the limit superior of $(a_n) := \limsup\ a_n$. Wanted to see if my proof made sense....
2
votes
1answer
50 views

Verify proof that $x_n = \sum_{k=1}^n {k \over 2^k}$ is bounded and find its supremum and infinum

The problem I'm solving states: Let $n\in \mathbb N$ and $x_n$ be a sequence: $$ x_n = \sum_{k=1}^n {k \over 2^k} $$ Prove $x_n$ is bounded and find $\sup\{x_n\}$ and $\inf\{x_n\}$ Let $S_n$ ...
1
vote
1answer
27 views

Computation of solution of minimization problem

I have $u(x) = e^x \in H^1 (\Omega)$ and $\Omega = (0,1)$ and should compute the solution $p \in \mathbb P^2$ of the minimization problem: $$|| u-p||_{H^1(\Omega)}^2 = inf ||u-q||_{H^1(\Omega)}^2$$ ...
1
vote
1answer
25 views

If $\dfrac{x}{y}\in A$ for $x,~y\in A,~x<y$ and $\sup A<1$, prove $\sup A\in A.$

Let $A$ be a non empty subset of positive reals, such that $\dfrac{x}{y}\in A$ whenever $x,~y\in A,~x<y.$ If $\sup A<1$, prove that $\sup A\in A.$ Attempt. If $\sup A\notin A$, then we could ...
2
votes
0answers
56 views

Infimum of the set

Prove that set $S= \{\sin(x)+2\sin(\sqrt{2}x)+3\sin(\sqrt{3} x) \}$ has infimum $-6$. x is positive integer I have proved that $-6$ is lower bound of this set, but I don't know how to prove that ...
0
votes
0answers
23 views

Prove that this $ \sup $ is finite

How can i prove that this supremum is not $ \infty $? $$ \sup_{\vert\xi\vert\geq 1} \left( \dfrac{\left(1 + \sqrt{4\vert\xi\vert ^{2}-1}\right)^{i}}{\vert\xi\vert ^{i-1}\sqrt{4\vert\xi\vert^{2}-1}} \...
2
votes
2answers
46 views

Infimum, Supremum,Minimum,Maximum of the set $S$

I am thinking about the following characteristics of the set - Infimum, Minimum, Supremum, Maximum, (Open/closed/neither/both). $S =\{9+\frac{(-1)^n}{2^n} ; n \in \Bbb{N}\}$ By writing the ...
1
vote
0answers
31 views

Conjugate function of negative logarithm: fault in my reasoning?

A problem, asked by Boyd and Vandenberghe (3.36f), is: Compute the conjugate function $f^*(y,u) = \sup_{x,t} \{ y^Tx + ut - f(x) \}$ of the negative generalized logarithm for second-order cone: $...
0
votes
0answers
23 views

Supremum of simple function

I am having troubles computing the following expression: $$\int_{\mathbb{R}} \sup_{x:f_X(x)} f_{Y|X}(y|x)dy$$ Now, my $X$ lies in the interval $[0,1]$ and basically my $Y$ is algorithmically chosen. ...
0
votes
0answers
22 views

Prove that m is a lower bound for S if and only if −m is an upper bound for −S.

Given : −S = {−s : s ∈ S}. Prove that m is a lower bound for S if and only if −m is an upper bound for −S. And Prove that if S is bounded below then its greatest lower bound satisfies inf S = − sup(...
0
votes
1answer
64 views

Any subset of $\mathbb{R}$ is a countable union of pairwise disjoint intervals.

Let $A\subseteq\mathbb{R}$ be non-empty and bounded. Also, $\forall x\in A$, $\exists\epsilon>0$ such that $(x-\epsilon,x+\epsilon)\subseteq A$. Prove that $A$ is a finite or countable union of ...
3
votes
2answers
58 views

Prove that $\sup(S-T)=\sup S-\inf T$

Let $S$ and $T$ be nonempty sets of real numbers and define $$S-T=\{s-t|s\in S,t\in T\}$$ Show that if S and T are bounded then $$\sup(S-T)=\sup S-\inf T\\ \inf(S-T)=\inf S-\sup T.$$ My proof: ...
0
votes
0answers
16 views

Exercise 7. Chapter 3. Barry Simon.

Why 3.1.35 and 3.1.36 implies 3.1.37? I have this: (I use $\sup(A)=c$ iff $\forall\epsilon>0\exists a\in A$: $c-a<\epsilon$) $\sup_{F\subset I} \left(\sum_{\alpha\in F} |\beta_{\alpha}|^2\...
1
vote
1answer
45 views

How to prove that infimum and limit inferior commute?

How is it possible to prove the following inequality? $\inf_{x \in [a,b]} \{\liminf_{n \to \infty} f_n(x)\} \le \liminf_{n \to \infty} \{\inf_{x \in [a,b]} f_n(x)\}$ I tried to solve this problem by ...
1
vote
1answer
50 views

Can't proove that a function is convex

I'm trying to proove that a couple of functions are convex but one of them is giving me a hard time. Here it comes : I can't see how this function can be written as the $\max$ or $\sup$ of some ...