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 ...
