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

learn more… | top users | synonyms (2)

1
vote
5answers
30 views

inf e sup of empty set?

If $S$ is an ordered set then the empty set is a subset of $S$. What are $\inf$ and $\sup$ of such set? To be honest I don't know what it could be. I'm not talking about of real numbers, but any ...
1
vote
2answers
27 views

If $A$ is infinite and bounded, the infimum of the difference set of $A$ is zero.

Let $A$ be a non-empty subset of $\mathbb{R}$. Define the difference set to be $A_d := \{b-a\;|\;a,b \in A \text{ and } a < b \}$ If $A$ is infinite and bounded then $\inf{A_d} = 0$. Since $a &...
-1
votes
1answer
33 views

help with real analysis [on hold]

Let $S \subset \mathbb{R}$ be nonempty. Show that if $u= \sup S$, then for every number $n$ belong to $\mathbb{N}$ the number $u -\frac{1}{n}$ is not an upper bound of $S$, but the number $u + \frac{1}...
1
vote
1answer
17 views

Relation between Supremum and limit superior

If $\sup\limits_{n\ge 1} a_n<\infty$ then obtains that $\limsup\limits_{n\to\infty} a_n<\infty.$ Can you explain that?
1
vote
0answers
80 views

Proof that $\sum_{i=1}^\infty 2^i (\mathbf{1}_{A_i})^*(x)$ is finite almost surely [on hold]

Let $(X, \mathcal{A}, m, T ) $ be a dynamical system and $f$ satisfying $\int |f| \ln^+ \ln^+ |f| {\rm d}m <\infty$. Put $A_i=\{x: 2^{i}\le f(x)< 2^{i+1}\}$ for $i\ge 2$ and $A_1= \{x:f(x)<4\...
1
vote
2answers
35 views

Prove that $\sup{A} = \inf{B}$ if and only if for each $\delta > 0$ there exists $x \in A$ and $y \in B$ such that $x + \delta > y$:

Let $A$ and $B$ be non-empty subsets of $\mathbb{R}$ such that for all $x \in A$ and $y \in B$ we have $x \leq y$. Prove that $\sup{A} = \inf{A}$ if and only if for each $\delta > 0$ there exists $...
3
votes
1answer
48 views

Is there a meaning to the notation “\arg \sup”?

When $f$ is a function on a set $A$, the notation: $\arg\max_{x\in A} f(x)$ denotes the set of elements of $A$ for which $f$ attains its maximum value. This set may be empty, for example, if $f(x)=x$ ...
2
votes
0answers
27 views

Exchanging supremum and conditional expectation

I've come across a problem which seems similar to this but quite different and can't find a way of going around it. I am working with a continuous process $Y_t$ generating the filtration $(F_t)$ and ...
1
vote
1answer
19 views

Doob's submartingale theorem

According to Doob's theorem, If $\{X_n,\mathcal{F}_n, n\in \mathbb{N}^* \}$ is a submartingale and $L_1$ - bounded, it means $$\sup\limits_{n\ge 1} \mathbb{E} (|X_n|)<\infty,$$ then $\{X_n\}$ ...
0
votes
4answers
49 views

How to find the supremum and infimum of this set?

$\{3-\frac{17}{n}:n\in\Bbb{N}\} \subseteq A \subseteq (-\infty,3].$ Prove that A has an upper bound and find $\sup(A) $.
0
votes
1answer
22 views

Proving the supremum of this set

Suppose we have a subset of reals A and a real number s exists such that for all natural n, s + 1/n is an upper bound for A while s - 1/n is not an upper bound for A. Apparently s = supA but I can't ...
1
vote
1answer
23 views

$\sup_{x \in \mathbb R} k^2e^{−kx^2}f(x)≤\sup_{x \in \mathbb R} (k+1)^2e^{−(k+1)x^2}f(x)$?

Assuming that $f$ is bounded, continuous, and non-negative, is it true that $$\sup_{x \in \mathbb R} k^2e^{−kx^2}f(x)≤\sup_{x \in \mathbb R} (k+1)^2e^{−(k+1)x^2}f(x)$$ I have a hard time proving this ...
4
votes
1answer
45 views

Abuse of notation for infimum and supremum

I would like to take the infimum and supremum of two sets $(\frac{1}{2} e^{8m+4} - 1, e^{8m+4} - 1)$ and $(\frac{1}{2} e^{8m+4}, \frac{3}{2}e^{8m+4})$, but writing $\sup((\frac{1}{2} e^{8m+4}, \...
0
votes
0answers
43 views

Supremum, max, min, inf of $S=\bigcup_{n \in \mathbb N}\left[\frac{1}{2^n},8-\frac{1}{n}\right)$

$S=\bigcup_{n \in \mathbb N}\left[\frac{1}{2^n},8-\frac{1}{n}\right)$ I wrote out the first few intervals and see that this converges to the open interval $(0,8)$. So given that I'm trying to find ...
0
votes
0answers
19 views

functional type equation

Let $f,g$ be two nonconstant positive functions on $I=[0,1]$ and we assume that : $$ \sup_{x\in I}\sqrt{f^2(x)+g^2(x)}=\sqrt{\sup_{x\in I}f^2(x)+\sup_{x\in I}g^2(x)} $$ This implies that $(f,g)$ are ...
0
votes
1answer
26 views

If $\sup_n \frac{n}{|b_n|}<\infty$ then $\limsup_{t\to\infty} \frac{\# \{n\ge 1:|b_n|\le t\}}{t}<\infty$

Let $\{b_n\}$ be a non-zero sequence of complex numbers, $n\in \mathbb{N}^*$. According to a artical, if $\sup_n \frac{n}{|b_n|}<\infty$ then $\limsup_{t\to\infty} \frac{\# \{n\ge 1:|b_n|\le t\}}{t}...
0
votes
2answers
21 views

Convergence to supremum using Squeeze Theorem and Limit Theorem?

Suppose that $B$ is a nonempty set of $ \mathbb{R} $ that is bounded above. Let b = sup($B$). Prove there is a sequence $b_n \in B$ that converges to b. So I know we can use the Squeeze Theorem, and ...
0
votes
1answer
35 views

Is sup of max, same as max of sup?

Let $\sigma_1, \sigma_2 \dots \sigma_n$ be functions of $\omega \in \mathbb{R}_+$. Is $\sup_{\omega}(\max_{i=1:n} (\sigma_i))$ same as $\max_{i=1:n}( \sup_{\omega}(\sigma_i))$? Could you also please ...
2
votes
3answers
47 views

Proving $-1$ is the infimum of $E=\{\frac{2}{n}+(-1)^n\mid n\in \mathbb{N}\}$

Prove: $\inf\{E\}=-1$ for $E=\{\frac{2}{n}+(-1)^n\mid n\in \mathbb{N}\}$ Let assume the contrary , there is $x\in E$ such that $\frac{2}{n}+(-1)^n\leq-1$. Because we are looking at negative ...
1
vote
1answer
52 views

$S := \{x \in \Bbb R^3: ||x||_2 = 1 \}$ and $T: S^2 \to \Bbb R$ is a continuous function. Is $T$ injective?

$S := \{x \in \Bbb R^3: ||x||_2 = 1 \} \subset (\Bbb R^3, || \cdot ||_2 )$ and $T:S^2 \to (\Bbb R, |\cdot |)$ is a continuous function. I've already shown that $$T_{\mathrm{max}} := \mathrm{sup}\{ T(...
0
votes
1answer
22 views

$A\subseteq R$ is an upper bounded set that contains at least two items/numbers. If $x < \sup{A}$, then $\sup{\left(A\setminus\{x\}\right)}=\sup A$

Prove: $A\subseteq \mathbb{R}$ is an upper bounded set that contains at least two items/numbers. If $x < \sup{A}$, then $\sup{\left(A\setminus\{x\}\right)}=\sup A$. My attempt: Since $A$ is upper ...
1
vote
2answers
26 views

Find the supremum and infimum

I have a set $E = \{x: x^2-x-1 < 0 \}$ for which I need to find the infimum and supremum (and minimum and maximum if exists). I'm not sure how to do it but after some calculation I cam up with $Inf(...
1
vote
1answer
69 views

What is meant by $sup(A\cup B )$

I am given two subsets $A,B$ of $\mathbb{R}$ which are not empty and are bounded above. Now according to a lemma, both of these sets have supremums. My issue is part of the question deals with sup$(...
0
votes
0answers
30 views

Supremum/Infimum if x in R^n

there is a function f(x)=(3+x^2)/(5+x^4). I should find the Supremum and Infimum BUT for x of R^n. I dont understand what x in R^n changes in this situation? I can see that the Supremum in this ...
1
vote
2answers
22 views

Supremum $\sup_{x\in (0,+\infty)}|F(x,t)|=\sup_{x\in (0,+\infty)}\dfrac{e^{-t^2}}{x+|t|}$

Let : \begin{aligned}F \colon (0,+\infty)\times \mathbb{R} &\longrightarrow \mathbb{R} \\(x,t) &\longmapsto F(x,t)=\dfrac{e^{-t^2}}{x+|t|}.\end{aligned} How they do to find $\...
0
votes
1answer
46 views

How can I find the sup, inf, min, and max of $\bigcup\left[\frac{1}{n}, 2-\frac{1}{n}\right]$

$$\bigcup\left[\frac{1}{n}, 2-\frac{1}{n}\right]$$ I'm not sure how to get started with this one. When I graph the two functions I see they intersect at the point $(1,1)$, which I take to be the ...
1
vote
0answers
16 views

Finding the upper derivitive of the compostion of two functions

Let $f$ be defined on $[a,b]$ and g a continuous function defined on $[\alpha , \beta ]$ that is differentiable at $\gamma \in (\alpha, \beta)$ with $g(\gamma)=c\in(a,b)$. Show that if $g'(\gamma)>...
0
votes
1answer
42 views

Inf and Supremum of $\{\arctan(x) \; : \; x \in \mathbb R\}$

I had trouble with this, but I think it's because there is no minimum and no maximum, correct? For the $\inf$ I got $-\frac{\pi}{2}$ and for the $\sup$ I got $\frac{\pi}{2}$ But on second thought I ...
1
vote
1answer
35 views

Verify sup, max, min, inf for $\{\frac{n+3}{n}\;:\;n \in \mathbb N\}$

I think I'm finally starting to understand this and wanted to check that I answered the following correctly: $\{\frac{n+3}{n}\;:\;n \in \mathbb N\}$ inf: 1 min: 4 sup: DNE max: DNE
1
vote
2answers
51 views

supremum, infimum, max and min - assistance understanding the difference

I think I understand the very basic concepts of these terms, but wanted to check my understanding here. The max is the largest number in the set. The supremum is the least upper bound number ...
0
votes
3answers
64 views

To prove $\sup B \leq \sup A$

Assume $A$ and $B$ are non empty and bounded above and satisfy $B \subseteq A$. Show that $\sup B \leq \sup A$ I am thinking of proving using contradiction, but I am getting nowhere. Someone please ...
0
votes
2answers
48 views

Show that $f(x) = \inf\{d(a,x) : a \in A \}$ is continuous

Let $A$ be a non-empty set in a metric space $(X,d)$. Define $f: X \to \mathbb{R}$ by $f(x) = \inf \{d(a,x) : a \in A \}$. Prove that $f$ is continuous. If $f$ is continuous, then $\forall \epsilon &...
0
votes
0answers
44 views

Hausdorff metric is an Ultra metric [duplicate]

Anybody prove the following statement. Hausdorff metric $H$ is ultrametric if $d$ is ultrametric. For any $A$ and $B$ closed and bounded subsets of $X$, $$H(A,B) = \max \{\sup \limits _{a \in A}...
-1
votes
1answer
29 views

Which of following inequality holds?

$f(x,y) = _{\theta,\phi}^{{sup}}$ $ ||e^{i\theta}x - e^{i\phi}y||_{2}^2$ ,where $x,y \in \mathbb{C^2}$ and $\theta,\phi \in \mathbb{R}$ $f(x,y) \leq ||x||^2 + ||y||^2 - 2Re (\langle x,y \rangle )$ ...
1
vote
0answers
21 views

Which of the inequality holds?

$f(x,y) = _{\theta,\phi}^{{sup}}$ $ ||e^{i\theta}x - e^{i\phi} y||_{2}$ ,where $x,y \in \mathbb{C^2}$ and $\theta,\phi \in \mathbb{R}$ Which of the following holds ? $f(x,y) \leq ||x||^2 + ||y||^2 ...
1
vote
2answers
33 views

Finding $\inf(S)$ and $\sup(S)$ for $S = \{\frac{(m+n)^2}{2^{mn}} : m,n \in \mathbb{N}\}$

Given a set S such that $S = \{\frac{(m+n)^2}{2^{mn}} : m,n \in \mathbb{N}\}$, what is $\sup{S}$ and what is $\inf{S}$? I think $\inf{S}$ is 0 because if we let $n = m$ and allow $m$ to approach ...
1
vote
1answer
31 views

Showing that a “convolution” operator is associative

I am dealing with the following operator $*$ : $(A*B)(t) = \inf\limits_{\tau\in\mathbb{R}} (A(\tau) + B(t-\tau))$. I would like to show that it is associative, i.e : $((A*B)*C)(t) = (A*(B*C))(t)$ . I'...
2
votes
2answers
51 views

Show that $\ker(T)=\{\varphi _n\mid\lambda_n\neq 0\}^\perp $

Let $T:H \to H$ be defined as $Tx=\sum_{n=1}^{\infty} \lambda_n \langle x,\varphi _n \rangle \varphi _n$, given that $\{\varphi _n\}_{n=1}^\infty$ is an orthonormal sequence (not necessarily a basis) ...
0
votes
0answers
31 views

Help finding the supremum and infimum of the subset.

For $A = $ {$0,1,2,3,4,5,7,8,9$} , let $B = ${{$2,3,4$},{$0,3,4,5,9$},{$0,2,3,8$}} $B$ is a subset of $P(A)$. Using the partial order ⊆ for $P(A)$ find inf$(B)$ and sup$(B)$. The first thing one ...
1
vote
1answer
42 views

$\inf_{x\in A}{\limsup_nd(x_n, x)} = \limsup_n[\inf_{x\in A}d(x_n, x)] $ for compact subset $A$.

let $ (X, d) $ be a complete metric space, $ A\subset X $ be compact and take a sequence $ (x_n) \subset X $\ $ A $ as a bounded sequence. Since infimum is independent from n , does the following ...
0
votes
1answer
20 views

Supremum property of a family of functions

Let $\{f_n\}_{n\in\mathbb{N}}$ be a family of functions from $\mathbb{R}$ to $\mathbb{R}$. a.) $\sup\{\sum_{n}f_n(x):x\in\mathbb{R}\}\leq \sum_{n}\sup\{f_n(x):x\in\mathbb{R}\}$. b.) $\sup\{\...
1
vote
2answers
21 views

Supremum property of functions

Let $f$ and $g$ be functions from $\mathbb{R}$ to $\mathbb{R}$. Prove that: a.) $\sup\{f(x)+g(x):x\in\mathbb{R}\}\leq \sup\{f(x):x\in\mathbb{R}\}+\sup\{g(x):x\in\mathbb{R}\}$ b.) $\sup\{f(x)+...
0
votes
1answer
31 views

Proof that if set A is contained in Set B, then the supremum of A is less than or equal to the supremum of B

If $A \subset B$, then $\sup_B \geq \sup_A$. $\textbf{Proof:}$ Let $k = \sup_B$. If $\sup_A$ were greater than $\sup_B$, then $\sup_A=k+\delta$ where $\delta$ is some positive number. Since, however,...
6
votes
1answer
77 views

Find the maximum and minimum values of $\sin^2\theta+\sin^2\phi$ when $\theta+\phi=\alpha$

Find the maximum and minimum values of $\sin^2\theta+\sin^2\phi$ when $\theta+\phi=\alpha$(a constant). $\theta+\phi=\alpha\implies\phi=\alpha-\theta$ $\sin^2\theta+\sin^2\phi=\sin^2\theta+\sin^2(\...
2
votes
0answers
38 views

How do I calculate the variation of a function?

I am trying to understand how to calculate the variation of a function. In this regard, the book that I am reading offers the following definition - $$V_g([a,b] = sup \sum_{i=0}^n |f(x_{i+1}) - f(...
1
vote
0answers
26 views

Hausdorff distance on power sets

Consider a general metric space $(S,d)$, with $d$ a $1$-bounded metric, and let $X,Y \subset S$ be two closed subsets of $S$. Notice that $X$ and $Y$ are not compact. Let $\mathcal{P}(X)$ denote the ...
0
votes
1answer
21 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
42 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 $d(x,y) = \|x-y\|$. ...
1
vote
3answers
59 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 $2\sin^...
1
vote
1answer
16 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 &= \...