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

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Find supremum and infimumm of a set with two variables

$$A= \left\{\frac{m}{n}+\frac{4n}{m}:m,n\in\mathbb{N}\right\}$$ Since $m,n\in \mathbb{N}$, infimum is zero because $m,n$ both are increasing to infinity. Then the supremum is $5$ when $m,n$ are ...
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16 views

$\liminf_n \min(a_n,b_n)=\min( \liminf_n a_n, \liminf_n b_n)$

Do you have a reference for the following intuitive result? Let $(a_n)_{n\ge 1}$ and $(b_n)_{n\ge 1}$ be two sequence of reals. Then $$\liminf_n \min(a_n,b_n)=\min( \liminf_n a_n, \liminf_n b_n).$$
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34 views

$\liminf_n a_n = \inf_n a_n$ if $a_n \ge a_m$ when $n\mid m$

I would like to ask a reference for the following very easy result: can someone help? Let $(a_n)_{n\ge 1}$ be a sequence of positive reals such that $a_m \le a_n$ whenever $n$ divides $m$. Then $$ ...
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1answer
12 views

Why is the lagrange dual function concave?

In a book I'm reading it says I'm struggling to understand the last sentence. Why can one conclude concavity from having a pointwise infimum of a family of affine functions?
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3answers
63 views

If $A \subset B$ m then $\inf B \geq \inf A $?

Suppose $A,B$ are nonempty sets such that $A \subseteq B$. I want to show that $\inf B \geq \inf A $. Suppose $x \in A$ arbitrary. We know $x \geq \inf A $. As $x \in B $, We know out previous ...
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2answers
24 views

What is the difference between maximal element and least upper bound?

Maximal element is given as: Let $(P,\leq)$ be a partially ordered set and $S\subset P$. Then $m\in S$ is a maximal element of $S$ if for all $s\in S$, $m \leq s$ implies $m = s$. Least upper ...
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2answers
63 views

Meaning of symbols like $\inf\limits_{\epsilon>0}$

I am very confused at the precise definition of the following symbols. A reference or explanation would be great. $$\Large\inf\limits_{\epsilon>0}\qquad \sup\limits_{\epsilon>0}$$
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1answer
45 views

$\sup(\sup) \leq 2 \inf(\sup)$

Consider a function $ F : [a; b] \times [a; b] \longrightarrow [0, \infty)$ such that $F(x; y) = F(y; x)$ and $F(x; y) \leq F(x; z) + F(z; y)$ for all $x, y, z \in [a; b].$ Prove that $$ \sup_{x \in ...
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1answer
34 views

A property of infimum??

Let $X$ be some space (eg. vector space or Banach space). When is it true that: for any $\epsilon >0$ small, there exists an $f \in X$ such that $$(1+\epsilon) \inf_{g \in X} I(g) \geq I(f)?$$ ...
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1answer
25 views

How does this follow from the theorem?[normed linear space]

I have this theorem: Let X and Y be normed linear spaces and let $T:X\rightarrow Y$ be a linear transformation. The following are equivalent: a. T is uniformly continuous. b. T is ...
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1answer
45 views

Prove that a sequence sup $a_n\ne1$

I have to prove the following: Let $(a_n)$ be a sequence such that $(a_n)<1$. Prove or disprove: If $(\frac{1} {1-a_n} )$ is bounded from above, then $\sup(a_n) \ne 1$. I was thinking ...
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2answers
59 views

If $\sup \{a_n\mid n\in \mathbb{N}\}=1$ then $\frac{1}{1-a_n}\to\infty$

Suppose $(a_n)$ is a sequence such that $a_n<1$ for all $n$ and $s:=\sup \{a_n\mid n\in \mathbb{N}\}=1$. I want to prove that $\frac{1}{1-a_n}\to\infty$. My initial approach: let $M>0$ and ...
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1answer
25 views

The supremum of the function $f(x)=\frac{1-\cos Nx}{1-\cos x}$

I have the following function: $f(x)=\dfrac{1-\cos Nx}{1-\cos x}$ Where N is integer. I know the function has Sup when x goes to $2n\pi$ $n\in\mathbb{N}$. But is it possible to show this? Thank ...
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1answer
17 views

Supremum over separable Banach space of a measurable function is measurable

Let $X$ be a separable Banach space. Suppose that $f:[0,T] \times X \to \mathbb{R}$ is such that $t \mapsto f(t,x)$ is measurable. Is the function $$t \mapsto \sup_{x \in X}f(t,x)$$ also measurable? ...
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2answers
60 views

Show that $\sup (A\cdot B)=\max\{\sup A\cdot\sup B, \sup A\cdot\inf B,\inf A\cdot\sup B,\inf A\cdot\inf B\}$

Given nonempty subsets $A$ and $B$ of positive real numbers, define $$A\cdot B=\{z=x\cdot y:x\in A,\,y\in B \}$$ show that if $A$ and $B$ are bounded sets of real numbers, then $$\sup(A\cdot ...
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1answer
73 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 ...
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1answer
51 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 ...
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0answers
16 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 ...
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0answers
21 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 ...
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47 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 ...
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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 ...
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1answer
16 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 ...
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2answers
43 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 ...
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4answers
95 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 ...
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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 ...
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1answer
33 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 ...
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1answer
64 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 ...
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1answer
33 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 ...
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1answer
30 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 = ...
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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 ...
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30 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 ...
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25 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 ...
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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 ...
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2answers
33 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?
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34 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 ...
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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 ...
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1answer
44 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 ...
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3answers
67 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 ...
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1answer
35 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 ...
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1answer
99 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: ...
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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 ...
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2answers
24 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 ...
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1answer
25 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 ...
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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
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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
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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
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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
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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
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2answers
42 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
53 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 ...