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

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3
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1answer
69 views

Boundedness of the function

Let $x\in(0,1)$ and $S_{n-1}=\sum\limits_{k=0}^{n-1}x^k$. Then define $f$ as the following :$$f(x)=\sum_{n=1}^{\infty}\left|\frac{nx^n}{S_{n-1}}-1\right|$$ I need to show that ...
2
votes
1answer
43 views

Function definition involving supremum and infimum

I am currently reading Vector Calculus, Linear Algebra, and Differential Forms by John Hubbard and Barbara Hubbard and am having a bit of trouble reconciling notation with definition. The book is ...
0
votes
0answers
15 views

Is this expression true for moments of random variables?

Suppose $X_1(t), \cdots, X_n(t)$ are random variables of a continuous time stochastic process. Suppose for any $p>1$, $\sup_{t \geq 0} E\left[\sum_{i=1}^n X_i(t)^p \right] < K_p$ where $K_p$ is ...
0
votes
0answers
19 views

finding the supremum

Let $A=\{x:\frac{[b\cdot n]}{n}\}$ when $n\in \mathbb{N}$ find the supremum we know that $b\cdot n-1<[b\cdot n]<b \cdot n$ therefore $b- \frac{1}{n}<\frac{[b\cdot n]}{n}<b$ so b is a ...
1
vote
2answers
45 views

Show that $\sup(\frac{1}{A})=\frac{1}{\inf A}$

Given nonempty set $A$ of positive real numbers, and define $$\frac{1}{A}=\left\{z=\frac{1}{x}:x\in A \right\}$$ Show that $$\sup\left(\frac{1}{A}\right)=\frac{1}{\inf A}$$ let ...
0
votes
0answers
12 views

Prove that the limit of $p$-th root of the integral of the $p$-th power of a function is it's supremum [duplicate]

I'm learning about $L^p$ spaces and it seems that $|| \cdot ||_\infty$ is sort of defined in such a way that's it's essentially the supremum of a function. I'm wondering if a simpler claim can be ...
0
votes
1answer
14 views

Infimum of the supremum absolute value of a decreasing sequence of subsets of $\mathbb{C}$ with non-empty intersection

Let $K_{n}$ be a decreasing sequence of bounded subsets of $\mathbb{C}$ such that $\cap_{n}K_{n}=K\neq\emptyset$. Let $\lambda_{n}=\text{sup }_{\lambda\in K_{n}}|\lambda|$ and $\lambda_{0}=\text{sup ...
0
votes
2answers
34 views

Supremum Definition

I am using this and am looking at problem 2. When looking at the solution for 2, it states that since $B - \epsilon < $sup $Y$ (and B is defined as sup Y), then there exists an element y in Y ...
5
votes
4answers
90 views

what is the sup and inf of absolute value of $1+z+z^2+ …+z^n$?

What is the supremum and the infimum of the absolute value of $1+z+z^2+ \dots+z^n$ when $z$ is a complex number and $z$ is inside the unit circle on the complex plane, which means $zz^*<1$? I ...
0
votes
1answer
27 views

Question about an extended linear functional on vector lattice of functions

I'm struggling with following problem: Let $\mathcal{F}$ be a vector lattice of bounded functions on a set $X$ such that $1\in\mathcal{F}$. Suppose that we are given a linear functional $L$ on ...
0
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0answers
31 views

Why should $\sup |\alpha_i - \beta_i|$ be real?

If $p,q$ are real polynomials such that $p = \sum_{i=0} ^{\infty}\alpha_i x^i$ and $q = \sum_{i=0} ^{\infty}\beta_i x^i$ where only a finite number of $\alpha_i ~'s, \beta_i~'s$ are zero, then why ...
0
votes
1answer
32 views

Linearity of the supremum

Short question: In which cases we have: Let $f_n$ be a sequence of functions, then $\sup_{k\in \mathbb N}(\sum_{n}^if_k(n))=\sum_{n}^i\sup_{k\in \mathbb N}f_k(n)$ ? I guess if the $f_k(n)$ are ...
3
votes
1answer
63 views

Simple question about the supremum of lebesgue integrals

Do we have the following equality? Let $h$ be measurable and non-negative, $f$ another measurable function and $g$ a step-function, then: $$\sup_g\left\{\int_X(fg)\,d\mu:0\leq g\leq ...
2
votes
1answer
32 views

Does there exist a sequence $\{I_n\}$ of intervals such that $I_{n+1} \subsetneq I_n , \forall n \ge 1$ and $\cap_{n=1}^\infty I_n=(0,1)$ ?

Does there exist a sequence $\{I_n\}$ of intervals such that $I_{n+1} \subsetneq I_n , \forall n \ge 1$ and $\cap_{n=1}^\infty I_n=(0,1)$ ? I know that not all the intervals can be closed , as ...
0
votes
1answer
27 views

Is there a counter-example to this problem on convergence of a bounded, strictly increasing sequence?

I came across this problem in an old set of class notes: Let $A \subset \mathbb{R}$ be a nonempty, bounded set. Let $\alpha = \sup{A}$, and let $(a_n)$ be a convergent sequence in $A$, with $a = ...
0
votes
1answer
27 views

Find $\liminf\limits_{n\to\infty} (x_{n})$ and $\limsup\limits_{n\to\infty}(x_{n})$ for a sequence $x_{n}=1-nsin\frac{n\pi}{4}$

Subsequence $a_{n_{1}}=sin\frac{n\pi}{4}$ is bounded ($[-1,1]$), and a subsequence $a_{n_{2}}=n$ is bounded below. We can find cluster points for the first subsequence $C_{1}=\{-1,1\}$. For the second ...
3
votes
0answers
27 views

Calculating the convex conjugate of the function $f(x)=\lim_{n\to \infty}\left(-\frac{1}{n}\log \sum_{k=1}^n e^{a_k\cdot x+b_k}\right)$.

The convex conjugate also known as Legendre–Fenchel transformation of a convex function $f:\mathbb{R}\to\mathbb{R}\cup\{+\infty\}$ is function $f^\ast:\mathbb{R}\to\mathbb{R}\cup\{+\infty\}$ definite ...
0
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0answers
19 views

ordinal number question

An ordinal i is a natural number if and only if every nonempty subset of i has a greatest element. If a set of ordinals X does not have a greatest element, then sup X is a limit ordinal can someone ...
2
votes
1answer
19 views

Upper bound for the infimum of $(K-x)^2 + (T/x)^2$

I have a function $$f(x)=(K-x)^2 + \left( \frac{T}{x} \right)^2$$ where $K$ and $T$ are positive constants and $x>0$. The function $f$ (hopefully) has an infimum, in terms of $K$ and $T$. I ...
1
vote
2answers
30 views

Norm is sup of inner products (proof).

Let $V$ be a vector space with an inner product $\langle.,. \rangle$ and associated norm $|| . ||$ Then: Could I have a proof of this fact?
0
votes
2answers
32 views

How to use the definition of supremum to prove that $\sup{S} = 2$?

Let $S$ $=$ {$2-\frac{3}{n+1} | n\in \mathbb{N}$}. Use the definition of supremum to prove that $\sup{S} = 2$. Here's what I have so far. We want to show that $2$ is an upper bound, and it is also ...
0
votes
1answer
25 views

Inequality with supremum of divergence of vector fields

I would like to write this inequality: $$\sup\left\{ \int_{A \cup (\mathbb{R}^d \setminus F)} \operatorname{div} T \space \rm d m^d : T \in C_0^1(\mathbb{R}^d,\mathbb{R}^d), |T(x)| \le 1 \right\} \le ...
0
votes
1answer
43 views

a question related to supremum and infimum

$T_n^*:=sup\{t|\sum ψ(x_i;t)\gt 0\}$ $T_n^{**}:=inf\{t|\sum ψ(x_i;t)\lt 0\}$ As it's seen in the above figure, $-\infty \lt T_n^{*} \le T_n^{**} \lt +\infty$ Then, how to write the two followings ...
1
vote
3answers
58 views

Supremum of a sequence of functions

I have a problem understanding the function $\sup f_n(x)$ on a domain of convergence. Given a sequence of functions $f_n(x)$ I guess that $\sup f_n(x)$ is the function such that for each x we find the ...
0
votes
1answer
33 views

Pointwise convergence of $c$-concave envelopes as the sequence of cost functions converges pointwise

On Remark 1.12, page 33 of Villani's Topics in optimal transport we find the following problem. It should be easy but it's turning me mad. Let $c:\mathbb{R}^n\times\mathbb{R}^n\rightarrow ...
3
votes
1answer
97 views

Greatest Lower Bound of the Set of Upper Bounds for a Function

I'm currently in the process of reading a paper, and am trying to work through some of the details on my own. In order to ask my question more effectively, I'm going to begin with a little background: ...
0
votes
1answer
29 views

Conditions for equality in the case of calculation of the supremum of $|f|^2$

It is obvious that $$\sup_{[0,T]}{\left(|f|^2\right)}\leq (\sup_{[0,T]}|f|)^2.$$ My questions is when does the equality hold? Progress: If $|f(x)|$ is not the zero function on $[0,T]$, the equality ...
1
vote
2answers
23 views

Image of a convergent sequence in an increasing function

Suppose I have a function $f:[a,b) \to \mathbb{R}$ that is increasing on its domain and a sequence $a_n \subset [a,b)$ such that $a_n \to b$ and $f(a_n)\to l\in \mathbb{R}$. How would I go about ...
0
votes
1answer
23 views

Prove: Monotonic And Bounded Sequence- Converges

Let $a_n$ be a monotonic and bounded sequence, WLOG let assume it is monotonic increasing. $a_n$ is bounded therefore there is a Supremum, $Sup(a_n)=a$, therefore $a_n<a+\epsilon$. On the other ...
0
votes
1answer
88 views

The importance of being real

Let $\Sigma$ be a collection of holomorphic, one-to-one function from some simply connected region $\Omega$, which map $\Omega$ into the open unit disc $U$. Fix $z_0 \in \Omega$ and put $$\eta = ...
1
vote
1answer
33 views

If the set $B=\{f(x) : x\in A\}$ has supremum and $C=\{k+f(x): x\in A\}$, then what is $\sup C$?

If the set $B=\{f(x) : x\in A\}$ has supremum and $C=\{k+f(x): x\in A\}$, then what is $\sup C$? Since $C$ is not an empty set and $f(x)\le \sup(B) ⇒ k+f(x)\le k+\sup(B)$. So $C$ is bounded ...
1
vote
2answers
27 views

Monotonous everywhere function

$f: \mathbb R \to \mathbb R,\forall x \in \mathbb R $ $\exists \delta \gt 0 : f$ is non-decreasing on $(x-\delta,x+\delta)$(I call that statement A). I need to prove that $f$ is non-decreasing on ...
3
votes
2answers
53 views

Sup and inf of $n \sin(1/n)$

If $n$ is a natural number then, what is the supremum and infimum of $n\sin(1/n)$? is the question I want to solve. I drew $sin(x)/x$ graph and I think that the supremum is $1$ and infimum is ...
2
votes
2answers
57 views

Calculate $\sup\limits_{x\in(0,+\infty)}\frac{x^2 e^{-n/x}}{n^2+x^2}$

Calculate $$\sup_{x\in(0,+\infty)}\frac{x^2 e^{-n/x}}{n^2+x^2}$$ The derivative is $$\frac{n e^{-n/x} (n+x)^2}{(n^2+x^2)^2}\geq 0$$ then $$\sup_{x\in(0,+\infty)}\frac{x^2 ...
0
votes
2answers
40 views

Limit of monotonic function

I have to prove that if $(x_1 \gt x_2) \Rightarrow (f(x_1) \ge f(x_2))$, then $$\forall a \in \mathbb R \exists L \in \mathbb R \lim_{x \to a^+}f(x) = L$$ I have a feeling that L = $inf_{x \in (a,a+ ...
1
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0answers
41 views

Let A= [0,1] - {1/n │n ∈ N}. Find sup(A), inf(A), min(A), max(A).

My idea of this question is to claim sup(A) and inf(A) exists (and equals a value) and prove by contradiction that min(A),max(A) exists afterwards (and equals sup(A),inf(A)). The issue that I have is ...
1
vote
0answers
42 views

Supremum/Infimum and min/max

(Problem written verbatim) Let A = [0,1] - {$\frac1n$|n $\in$ $\mathbb{N}$}. Find supremum, infimum, maximum, and minimum of A. Justify your answers. I'm not really sure where to start. I tried ...
2
votes
2answers
24 views

Supremums and subsets

Let $A \subseteq B$, where $A$ and $B$ are non-empty sets and $B$ is bounded above. Show that $\sup(A)$ exists and $\sup(A)\le \sup(B)$ (Problem given as written) For clarity, $\sup(x)$ is the ...
0
votes
0answers
23 views

Find sum of supremum and infimum of $\tan(x)^{\sin 2x}$

Given a function $f(x) = \tan(x)^{\sin 2x}$ I have to show that it has infimum and supremum, this was easy. I already proved that $f$ takes the value equal to infimum (respectively supremum) for ...
1
vote
2answers
43 views

What exactly is the meaning of the following $\inf\{ s_n : n > N\}$ and $\sup\{ s_n : n > N\}$

What exactly is the meaning of the following $$u_N = \inf\{ s_n : n > N\} \ \ \text{ and} \ \ v_N = \sup\{ s_n : n > N\}$$ This might seem a stupid question, but I am not understanding the ...
-1
votes
1answer
49 views

Existence of a sequence that converges to infimum of a function

X is a compact subset of $\Re^{n}$. A upper and lower bounded function f is defined that f:X $\rightarrow$ $\Re$. Does a sequence of {$x_n$} always exist so that f($x_n$)$\rightarrow$ $\inf$[f(x)]. ...
0
votes
1answer
88 views

Equality of sets supremum and infimum

Suppose we have a right-continuous function $f$ defined on $[0,\infty)$. I now would like to prove if $\lambda >0$ and $t\geq 0$ we have that \begin{equation} \left\{ \inf \{ s\geq 0 : f(s) \geq ...
2
votes
2answers
62 views

${\rm sup}\ A\cap B = {\rm min}\ \{ {\rm sup} (A), {\rm sup}(B) \} $

Let $A,B\subseteq \mathbb{R}$ be a non-empty intervals and bounded from above. If $A\cap B\neq \emptyset $ prove that it is bounded from above and that $Sup(A\cap B)=min\{sup(A),Sup(B)\}$ ...
0
votes
1answer
56 views

If $A$ has no max and $B$ is finite, then $\sup(A)=\sup(A\setminus B)$

Let $A\subset \mathbb{R}$ be non-empty and bounded from above, and assume it does not have a maximum. Let $B$ be a finite set of real numbers. Prove: $\sup(A)=\sup(A\setminus B)$ ...
1
vote
1answer
24 views

Prove the least upper bound of a set is in the closure of that set

If $S \subset \mathbb{R}$ and $x$ is the least upper bound of $S$, prove $x$ is in the closure of $S$. I think this means I either have to show that $x \in S$ or $x$ is a limit point of $S$, so I ...
1
vote
2answers
55 views

Prove $\sup(A)=\sup(B)$

Let there be $A,B\in \mathbb{R}$ non-empty, let assume that $\forall a\in A \exists b \in B$ such that $a\leq b$ and $\forall b\in B \exists a \in A$ such that $b\leq a$. Show that $A$ is ...
0
votes
3answers
41 views

Prove that a number $u$ is $\sup S$ given certain properties.

Problem Let $S$ be a nonempty subset of $\mathbb{R}$, and let $u$ be a number with the following properties: for each positive integer $n$, the number $u - \frac{1}{n}$ is not an upper bound of ...
0
votes
1answer
10 views

Supremums of sequences

Let $x_n$ and $y_n$ be two sequences of real numbers. Assume that $y_n$ is bounded above and that $x_n$$<$$y_n$ for all n$\in$$\mathbb{N}$. (a) Prove that $x_n$ is also bounded above. (b) Prove ...
0
votes
2answers
20 views

Proving supremums of sequences

If $l$=sup($x_n$), what is sup($kx_n$) where k$\in$$\mathbb{R}^{+}$? Prove your conjecture. I have that sup($kx_n$)=$kl$. I can prove that it is an upper bound of $kx_n$, but I'm having trouble ...
0
votes
1answer
25 views

condition for a supremum

let $A\subseteq \mathbb{R}$ be a non-empty set and $s\in \mathbb{R}$ and upper bound of $A$. So $s$ is the supremum of $A$ $\iff$ $\forall \epsilon>0$ there is $x\in A$ so $s-\epsilon<x\leq s$. ...