For questions on suprema and infima. Use together with a subject area tag, such as (real-analysis) or (order-theory).

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1answer
14 views

Prove that $M(|f|,S)-m(|f|,S)\leq M(f,S)-m(f,S)$

Let $M(f,A)=sup\{f(x):x\in A\subseteq[a,b]\}$ and let $m(f,A)=inf\{f(x):x\in A\subseteq[a,b]\}$. Given that $|f(x_{0})|-|f(y_{0})|\leq |f(x_{0}-f(y_{0})|\leq M(f,S)-m(f,S)$ for $x_0,y_0\in S$, prove ...
3
votes
1answer
28 views

Infimum of distance between point and (closed) set

I'm having a little trouble with the following exercise: Let $V \subset\mathbb{R}^p$ be a non-empty, closed set and $a \in \mathbb{R}^p$. For $x, y \in \mathbb{R}^p$ we note $d(x,y) = \|x-y\|$. ...
0
votes
3answers
55 views

Find the maximum and minimum values of $x-\sin2x+\frac{1}{3}\sin 3x$ in $[-\pi,\pi]$

Find the maximum and minimum values of $x-\sin2x+\frac{1}{3}\sin 3x$ in $[-\pi,\pi]$. Let $f(x)=x-\sin2x+\frac{1}{3}\sin 3x$ $f'(x)=1-2\cos2x+\cos3x$ Put $f'(x)=0$ $1-2\cos2x+\cos3x=0$ gives ...
1
vote
1answer
14 views

Infimum inequality comparing restrictions.

Suppose $f$ is a continuous function on the real line. Say we have two collections of sets $\{A_k\}_{k=1}^{n}$ and $\{B_k\}_{k=1}^{m}$, where $n>m$ and \begin{align} \bigcap_{k=1}^{n} A_k &= ...
0
votes
2answers
32 views

Approximating $\inf\sup$

Let $f\colon A \times B \rightarrow \mathbb{R}$. I am reading a paper in which some quantity of the form $$ \inf_{a\in A}\sup_{b\in B}f(a,b) $$ is equal to $$ \inf_{a\in A^{\prime}}\sup_{b\in ...
5
votes
1answer
31 views

Expanding a norm over a given space

Let $1 \leq p<q \leq \infty$ (p an q are not related) Let $\Phi$ be the vector space of all sequences with at most finitely many nonzero elements, meaning $\Phi=\{\{x_n\}_{n=1}^\infty|$ there is ...
0
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0answers
20 views

Solution to supremum inequality

I am currently learning for an exam. The task is to find a $k$ that fulfills the following condition: $$\sup_{\omega\in [-\pi,\pi]} \lVert 1 - G_c(z)L(z) \rVert _{z=e^{i\omega}} < 1$$ where ...
0
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0answers
16 views

Inequality with supremum of a specific integral

I'm reading Kai Diethelm's "The analysis of fractional differential equations" and there's a part I haven't been able to figure out. It's from page 93. Let $p >0.$ Why does the following ...
0
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0answers
14 views

special matrix name (similar to cyclic matrix)

does the matrix that follows the following condition have a special name? Setting $$A=(a_{ij})$$ $$a_{ij} ∈ R$$ Notation $$A^2=A*A=(^2a_{ij})$$ $$A^n=(^na_{ij})$$ condition $$ \forall ...
2
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1answer
20 views

Showing uniform convergence to origin in 3rd quadrant for $x(t)=\frac{1}{\frac{1}{x_0}-t}$ as $t\ \rightarrow \infty$

I want to show that for the system $\dot{x}=x^2, \dot{y}=y^2$,any solutions starting in the 3rd quadrant not including 0, converge uniformly to the origin. For an initial point $(x_0,y_0)$, (note both ...
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2answers
25 views

Matrix norm inequality proof - does this use Cauchy-Schwarz?

The matrix norm for $A : \mathbb{R}^n \rightarrow \mathbb{R}^m$ (so $A$ is an $m \times n$ matrix) is given by $$\|A\| = \sup_{X \in \mathbb{R}^n \setminus \{0\}} \frac{|AX|}{|X|}$$ where $| \cdot |$ ...
2
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2answers
78 views

How to properly find supremum of a function $f(x,y,z)$ on a cube $[0,1]^3$?

Solving an applied problem I was faced with the need to find supremum of the following function $$f(x,y,z)=\frac{(x-xyz)(y-xyz)(z-xyz)}{(1-xyz)^3}$$ where $f\colon\ [0,1]^3\backslash\{(1,1,1)\} ...
-2
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1answer
21 views

Finding min, max, inf, sup for $\frac{m*n}{4m^2+n^2}; m\in \mathbb{Z}, n \in \mathbb{N}$ [closed]

What is the proper way to find $\inf, \sup, \max, \min$ for: $A=\{\frac{m*n}{4m^2+n^2}; m\in \mathbb{Z}, n \in \mathbb{N}\}$ I'm not sure what is the proper proof outline for this case. Thanks!
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0answers
25 views

Find supremum, infimum for $A=\{n+ \frac{x}{n} : x \in R \setminus Q, n \in N, n \le \sqrt 5, |x| \lt \sqrt5\}$

Given expression: $A=\{n+ \frac{x}{n} : x \in \mathbb{R} \setminus \mathbb{Q}, n \in \mathbb{N}, n \le \sqrt 5, |x| \lt \sqrt5\}, \mathbb{N}\setminus0 $ My goal is to find out $\sup(A)$, $\inf(A)$, ...
2
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2answers
35 views

Showing that $\sup\{|f(x)-f(y)|, x,y\in X\}= \sup \ f - \inf \ f$

I need to show that, for $f:X\to \mathbb{R}$ bounded, we have: $$\sup\{|f(x)-f(y)|, x,y\in X\}= \sup f - \inf f$$ Well, I know that $$\sup\{|f(x)-f(y)|, x,y\in X\}\ge |f(x)-f(y)|$$ but in what ...
0
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1answer
45 views

sup and inf of $\{\frac{(-1)^n}{n}, n\in \mathbb{N}\}$

I need to prove that $\sup A = \frac{1}{2}$ and $\inf A = -1$ for: $$A = \{\frac{(-1)^n}{n}, n\in \mathbb{N}\}$$ Well, for $n\in 2\mathbb{N}$, we have: $$A' = \{\frac{1}{n}, n\in 2\mathbb{N}\} = ...
0
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0answers
27 views

Proving $\sup A = -\inf(-A)$ [duplicate]

In order to prove that $$\sup A = -\inf(-A)$$ obs: $A$ is limited I did: think of $\sup A$ as a number $$a\le \sup A, \forall a\in A \implies -a\ge -\sup A, \forall a\in A$$ Doesn't that implies ...
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2answers
48 views

Let $A = (0,1]$. Then $\inf(A) = 0$

I am having a problem with this statement. I am trying to prove that 0 is the greatest lower bound by showing that every lower bound greater than 0 is a contradiction but I can't figure our how. ...
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1answer
28 views

Supremum calcualtion, how do I get the answer [closed]

Find the supremum of G={x|(x^2+1)^−1>1/2} Could someone help me out with this calculation? Thanks. Appreciate. Options are: ...
0
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1answer
24 views

$\sup\limits_{t>0}[\frac{g(t)}{\sup\limits_{t<u<\infty}g(u)}]\leq 1$

I can't prove that the following inequality true or not: $$ \sup\limits_{t>0}[\frac{g(t)}{\sup\limits_{t<u<\infty}g(u)}]\leq 1 \tag{*}, $$ where $g$ is a positive function. I think it is true ...
0
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1answer
26 views

Let B be set of all twice differentialbe function $ f(0)=1, f'(0)=-1$ . .. Find supremum of $ {(f''(0):f\in B})$

Let B be set of all twice differentiable function $f$ such that $f: (-1,1) \to (0,\infty)$ and $ f(0)=1, f'(0)=-1$ . We have new function $g(x)$ such that $g(x)=\frac{1}{f(x)}$ and $g(x)$ is convex ...
0
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0answers
30 views

Banach space and invertible linear operator

Let $X$ be a Banach space. Let $T : X \to X$ be a invertible linear operator and $M > 0$ be such that $\|T^{-k}\| \le M$ for all $k \ge 1$. Prove that $\inf_ {n\ge1} \|T^n(x)\| > 0$ for all $x ...
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1answer
41 views

how do I find supremum of the set [closed]

I have a test on Monday and the professor gave a hint about the problem. f : ??? is random How do I solve this problem? Let $f: [-10, 10] \to \mathbb{R}$ be defined by $f(x) = ???$. Let $\Delta$ ...
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0answers
41 views

Support function of a set: when we can replace the supremum with the maximum?

Consider a set $K\subseteq \mathbb{R}^d$ (not necessarily closed and/or convex). Let $k:=(k_1,...,k_i,...,k_d)$ be a generic element of $K$. Consider the function $h_K:\mathbb{R}^d\rightarrow ...
0
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1answer
37 views

Why the supremum of a closed set is not necessarily equal to the maximum?

Consider a partially ordered set $A\subseteq \mathbb{R}^k$. I know that if $A$ is compact then $\sup A=\max A$. Could you give an intuition of why having $A$ closed is not enough to conclude $\sup ...
1
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1answer
15 views

Inequality about absolute value of difference of supremums

Let $A$ and $B$ be nonempty bounded subsets of $\mathbb{R}$. Define $|A-B|=\{|x-y|: x\in A, y\in B\}$. I wish to prove the following claim. Claim: $$|\sup A-\sup B|\leq \sup |A-B|$$ Is the following ...
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0answers
22 views

Sufficient conditions to have the supremum of a continuous function continuous?

Consider a function $f:\mathcal{X}\times \mathcal{Y}\rightarrow \mathbb{R}$ with $\mathcal{X}\subseteq \mathbb{R}^k$ and $\mathcal{Y}\subseteq \mathbb{R}^p$. Under which sets of conditions is ...
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0answers
26 views

Any regulated function on $[a, b]$ is bounded - proof verification

I would like to verify that my proof on the following statement is correct: Any regulated function on $[a, b]$ is bounded. Let $f \in R[a, b]$ (the space of regulated functions on $[a, b]$) be ...
0
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1answer
26 views

Writing $\sup$ and $\inf$ in unions

Could someone please explain why the following hold for a set of a functions $f_n(x)$: $$\{x : \sup f_n > c \} = \bigcup_{n=1}^\infty \{ x: f_n(x) > c \}$$ $$\{x: \inf f_n < c \} = ...
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2answers
59 views

$s \lt t$ for each $s \in S$ and each $t \in T$. Prove that $\sup S \le \inf T$ [closed]

Let $S$ and $T$ be subsets of $\mathbb R$ such that $s \lt t$ for each $s \in S$ and each $t \in T$. Prove carefully that $$\sup S \le \inf T$$ Best way to prove such a question?
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3answers
33 views

Is this true :if $x\in [0.2] $ then $f(x)=\frac{2x+3}{x+2} \in [0.2]$?

I'm sorry to ask this question mayeb it's a trivial question but i would like to confirme if i have this function $f(x)=\frac{2x+3}{x+2}$ which $x$ is a real number in $[0.2] $ then $f(x) \in [0.2]$ ...
0
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1answer
37 views

If S = {1/n | n ∈ ℕ}, what is inf(S)? [duplicate]

I believe the answer is 0 but I'm not really sure how to prove it...does it involve using epsilon?
0
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3answers
31 views

Real analysis Supremum and Infinum

Let S and T be subsets of R such that s < t for each s ∈ S and each t ∈ T. Prove carefully that sup S ≤ inf T. This question has been posted a few times, but i don't think the answers are formal ...
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2answers
34 views

$E$ bounded above, show that there is a sequence $(a_n) \subset E$ such that $a_n \to \sup E$

Let $E$ be an infinite set of real numbers which is bounded above. Show that there exists a sequence $( a_n ) \subset E$ such that $\lim a_n = \sup E$ if $\sup E \not \in E$. My attempt: Let ...
0
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1answer
17 views

Infimum and supremum of function

Let $a >0$ and consider function $g : \mathbb{R} \rightarrow \mathbb{R}, g(x) = x e^{-ax^2}$. Find infimum and supremum of $g$.$\lim_{a \to 0} x e^{-a x^2} = x$, so we can obtain every value of ...
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0answers
28 views

Prove the supremum = the least upper bound and the infimum = the greatest lower bound

Consider N (natural numbers) with a partial ordering of "divides", i.e. define a <= b if and only if a|b. For a,b ∈ N, let the gcd(a,b) denote the greatest common divisor of a and b, and let ...
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0answers
15 views

Infimum and supremum of (A*B)

I have been trying to solve this task for a while but I can't get it done. The task is: Let $A$ and $B$ be two bounded sets. If AB={ab| a,b>0, aEA, bEB} prove that, (a) $\inf(AB)=\inf(A)\inf(B)$ ...
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0answers
22 views

Closed operators

I was wondering whether the following statement is true or not? If $A$ is closed, then it follows from the closed-graph theorem that it is bounded iff $D(A)$ is closed. I found this in a chapter of ...
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2answers
29 views

If $f$ is continuous on $[a,b]$, then $f$ is bounded on $[a,b]$. Have questions about the proof

If $f$ is continuous on $[a,b]$, then $f$ is bounded on $[a,b]$. First off, I have seen a proof of this and I sort of get it but wanted to go through the proof and justify my arguments correctly to ...
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1answer
17 views

Is infimum of $\left\{\dfrac{1}{2} \sum_{n=1}^\infty \alpha_n \left( \dfrac{<u,e_n>}{\alpha_n} - <x,e_n> \right)^2 \right\}=0$?

Let $H$ be a Hilbert space, $\{e_n\}_{n=1}^\infty$ be an orthonormal basis for $H$ and $\{\alpha_n\}_{n=1}^\infty$ be a sequence in $(0,1)$. Let $u\in H$. Are we sure that $$\inf_{x\in H} ...
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3answers
27 views

I'm having trouble understanding this proof regarding Supremum.

I am trying to understand this proof, I understand the general form of the argument, and I understand what they are trying to show, Im just having trouble following the steps they've used. if anyone ...
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0answers
13 views

alternative proof that a function is homogeneous of degree one

Given a (profit) function of the form $$ \pi(p) = \sup \{p.y:y \in Y\} $$, where $p \in R_+^k$ is a positive (price) vector and $Y \in R^k$ is a (production-possibility) set. I need to proof that ...
1
vote
1answer
29 views

Existence of an increasing subsequence of sets that preserves supremum

Background This is not necessary to understand the question The following corresponds to a claim made on page 7 of the book Optimal Stopping and Free-Boundary Problems, Peskir and Shiryaev 2006. ...
0
votes
1answer
20 views

Infimum of $\left|\frac{p+q+p^{-1}+q^{-1}}{\ln(p)-\ln(q)}\right|$

Let $A$ be a positive constant and $\textbf{D}$ the set $\left\{(p,q)\in\textbf{R}^2 \mid 0<q<p \right\}$. I am looking at the infimum of the expression ...
1
vote
3answers
28 views

Can a bounded function always attain its least upper bound on a bounded rectangle in $R^n$? [closed]

Suppose we have a rectangle $Q$, and $Q\subset R^n$. Then $Q$ is bounded by the definition of higher dimensional rectangles. Suppose $f$ is a bounded function defined on $Q$. Since $f$ is bounded, we ...
0
votes
1answer
28 views

Supremum of a supermartingale

For a nonnegative supermartingale $(M_n)_{n\in\mathbb{N}}$ I would like to prove that $$E[\sup_n M_n] \leq E[M_0]$$ I have that $E[M_n] \leq E[M_0]$ for every $n$. This combined with nonnegativity ...
1
vote
1answer
27 views

$\sup_{x \in \mathbb{R}, x \neq 0} \frac{\|Ax\|}{\|x\|}$ equivalent to $\sup_{\|x\| = 1} \|Ax\|$ [duplicate]

I have seen it mentioned in many places that for some matrix $A \in \mathbb{R}^{n \times n}$ $\displaystyle\sup_{x \in \mathbb{R}^n, x \neq 0} \frac{\|Ax\|}{\|x\|}$ is equivalent to ...
1
vote
1answer
53 views

Supremum of expectation.

Can everyone help me prove the following inequality?? $$\sup_n\mathbb{E} {[|X_n|^2]}\le \left( \sup_n\mathbb{E} {[|X_n|^p]}\right)^{2/p}, $$ with $\{X_n\}$ is a centered independent sequence and ...
0
votes
1answer
24 views

Proof verification regarding supremum of a set

I would like to get some assistance with verifying my proof The point was to prove that set A has a supremum and find this supremum. is it a legitimate proof ? especially the part when I reach ...
0
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0answers
39 views

Proving a characterization for the existence of a supremum in a sublattice of the powerset lattice.

Suppose you have a class $\mathcal{C}$ over some set $A$, closed under intersection and union. This class forms a lattice with inclusion as the order relation, and union and intersection as join and ...