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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).

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Proof that Sup$(\{\frac{n}{n+1}| n \in \mathbb N \})=1$

I am practising for an exam and wish to show: Sup$\Big(\Big\{\frac{n}{n+1}\,\Big|\, n \in \mathbb N \Big\}\Big)=1$ We usually start out with noticing that for all $n \in \mathbb N$: $$ \frac{n}{n+...
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1answer
47 views

Antiderivative Supremum inequality $F(b)-F(a) \le (b-a)\sup\{f(x):x \in [a,b]\}$

Hello everyone I am suppose to show the following: Let F be the antiderivative of f, show that $$ F(b)-F(a) \le \sup\{f(x):x \in [a,b]\} $$ I figured I could use the Mean-Value-Theorem since $F'(...
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27 views

inequality with differential norm

I'm trying to prove that $\sup_{x,y \in B}\frac{|f(x)-f(y)|}{|x-y|} = \sup_{x \in B} \|D_f(x)\|$ Where $f:B \to \mathbb R^n$ is differentiable and $B \subset \mathbb R^n$. I have difficulties. this ...
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1answer
16 views

Supremum of the image of a monotonic function

For a monotonic function $f:D\to\mathbb{R}$ where $D\subset\mathbb{R}$, and two sets $A$ and $B$ such that $\sup A=\sup B$, is it true that $\sup f(A)=\sup f(B)$, where $f(A)$ denotes the image of set ...
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36 views

If $x$ and $y$ are arbitrary real numbers with $x < y$, prove that there exists at least one real satisfying $x < z < y$

This is a question from Apostol Calculus section I.3.12 q1 If $x$ and $y$ are arbitrary real numbers with $x < y$, prove that there exists at least one real satisfying $x < z < y$ I've ...
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29 views

What does Arg {Inf I(d)} means?

I am currently studying the phase-field method for fracture modeling. In an article by Miehe -"Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field FE ...
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1answer
44 views

Expectation of the supremum of a sequence of random variables

Let $\Omega = [0,1]$, $\mathcal{F} = \mathcal{B}(0,1)$, P=Lebesgue measure. Let $X_n(w)= \begin{cases} 0 \quad \frac{1}{n} < w \leq 1 \\ n-n^2w \quad 0 \leq w \leq \frac{1}{n} \end{cases}$ The ...
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1answer
32 views

Confusion on Additive Property of Supremum and Infimum - Theorem I.33 Apostol Calculus I

There is a great question and answer to the first part of Theorem I.33 of Apostol Calculus "Additive Property" here. I'm hoping someone can verify my attempt at part (b). Apostol provides the ...
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26 views

Supremum of a set and supremum of a sequence proof.

Let $A$ and $B$ be two nonempty bounded subsets of $\mathbb{R}$ with the following property: For all $b \in B$ there is a sequence $(a_n)$ of elements in $A$ such that $a_n \to b$ as $n \to \infty$. ...
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1answer
28 views

Compact set always contains its sup and inf

I am sorry if my question might seem minor, but I just want to clarify this point for myself. So in other thread on mathstackexchange, it is stated and proved that compact sets always contain their ...
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1answer
51 views

Determine $\sup \{xy−x^2/2\mid x\in [-1,1]\}$

I am trying to find the supremum of the following set $\{xy−x^2/2\mid x\in [-1,1]\}$, where $y$ is a real number. I am not sure if this is correct, but I managed to find that $$ \sup \{ xy-x^2/2\mid x\...
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2answers
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explanation of E[X] = Sup(E[Y] : Y a simple r.v.)

Could someone explain me the meaning of the following expected value of a positive random variables $X$? $\mathbb E[X] = \sup(\{\mathbb E[Y] : Y\text{ a simple r.v. with }0 < Y < X\})$ where ...
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1answer
11 views

Bounded from below implies the existence of infimum?

There exists proposition that says that every sequence bounded from above admits a supremum. Is it also correct to say that every sequence bounded from below admits an infimum? Tks
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1answer
24 views

Showing that $\sup(A\cdot B) = \sup(A)\cdot\sup(B)$ for $A$ and $B$ subsets of non-negative reals

While $A \cdot B=\{x \cdot y \mid x \in A, y \in B\}$, show that, for $A$, $B \subseteq [0,\infty)$, $$\sup(A \cdot B)= \sup(A) \cdot \sup(B)$$ My demonstration: First: $$\forall a\in A, a\le\...
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If $p(x)<1,$ then why does there exists $\alpha \in (0,1)$ such that $\alpha^{-1}x\in M$?

Definition: Given that $E$ is a normed linear space. Let $M\subseteq E$ be an open, convex set with $0\in M.$ For all $x\in E$, define \begin{align} p(x)=\inf\{\alpha>0:\,\alpha^{-1}x\in M \} \end{...
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1answer
47 views

Legendre transformation of a piecewise function

Let $f:\mathbb{R}\to\mathbb{R}$ be defined by $$f(x)=\begin{cases} 1 & \text{ if } x<1 \\ 3x-1 & \text{ if } x\geq 1 \end{cases}$$ How do I determine the Legendre transform $f^*$ of $f$, ...
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1answer
24 views

Continuity of an upper envelope

I'm now approaching for the first time to the definition of upper envelope of a family of functions, and I just wonder to know some basic properties. Suppose $\Omega $ is an open subset of $\mathbb{R}...
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1answer
26 views

Proving that a supremum is unique by contradiction

I am not sure if my proof is proper, so please comment on it and try to fix it if you can. Let $S$ be a non-empty set. Say that a and b are both supremum of $S$ with $a<b$. Assuming the ...
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Question about the completeness axiom

I have been looking online and on lecure notes and I have observed that there are 2 definitions for the completeness axiom and I cannot relate them together. These are: 1) Every non-empty set of ...
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3answers
42 views

Infimum and absolute values (Rudin's baby analysis)

In the second last inequality of the proof of Theorem 6.17 (Rudin's baby analysis), I think he uses the fact that $$|x-y|\leq c \text{ implies } |\inf x-\inf y|\leq c,$$ where $c$ is a real constant ...
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1answer
62 views

Infinum of exponential function

Assume that $x > c > 0 $. Prove that $\displaystyle \inf\left(c/x \frac{e^{c/x}}{e^{c/x} - 1}\right) = 1$ without using limit. However you are all allowed to use the theorem about the ...
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2answers
60 views

Does $\sup|A-B| \ge |\sup A -\sup B|$?

I have strong intuition that "yes" is the answer, because the difference between any two points between $A$ and $B$ can be only bigger (or equal) to the difference between two specific points. But I ...
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2answers
60 views

Infimum of the set $\{x\in \mathbb{Q}\;|\;x^2<2\}$?

My first year Analysis textbook at university includes examples to grasp the concepts of infimum, supremum, maximum, minimum, lower bound and upper bound in set theory for subsets of the real numbers $...
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2answers
54 views

Limits of monotone function

What does the notation $f(x^{+})$ and $f(x_+)$ mean? The context is the following I have a proposition concerning monotonic increasing functions, so $f$ is nondecreasing, also $x\in(a,b) =I$ where $f$...
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2answers
37 views

Let $B,C$ be sets, $B\subseteq C$ such that $\forall c∈C,\exists b\in B: c\le b$ then $\sup B=\sup C$.

I had been reading this. In the proof, below lemma is used. I don't know how to go for proving it.Notice that I want to prove this theorem for set of ordinals not real numbers Let $B,C$ be sets, $...
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1answer
44 views

$f: [a,b]\to\Bbb R $. If $g(x)=\sup\{f(t):t∈[a,x]\}$ and $f$ is continuous , then prove that $g$ is continuous at $a$

$f: [a,b]\to\Bbb R $. If $g(x)=\sup\{f(t):t∈[a,x]\}$ and $f$ is continuous , then prove that $g$ is continuous at $a$. My answer : Because $f$ is continuous in $[a,b], f$ is continuous at $a$. So, ...
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1answer
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Part of proof that supremum norm is (positive) definite. $||f||_\infty =0 \implies f=0$.

I want to prove that for a bounded function $f$, we have: $$ ||f||_\infty =0 \implies f=0. $$ Simply observe that if we denote the domain of definition by $D$: $$ ||f||_\infty = \sup_{x \in D}|f(x)...
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2answers
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getting the area of a set using the supremum and infinum concept

Prove that the area between the $x$-axis and the function $y=e^x$ in the interval $0 < x < c$ is $e^c - 1$. You're not allowed to use integrals. I have started to calculate the area by using ...
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1answer
37 views

Show that $\inf\left\{b/a:b\in B,a\in A\right\}=\inf{B}/\sup{A}$

Let $ A $, $ B $ two subsets of the real line; let, for at least one $ a>0 $, $ a\in A $. I've tried to prove that $ \inf\left(B/A_{>0}\right)=\inf{B}/\sup{A} $, were $ A_{>0} $ is meant to ...
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2answers
60 views

Find the subsequential limits for $\{x_n\}=\left\{1,{1\over 10},{2\over 10},\cdots,{9\over 10},{1\over 10^2},\cdots{10^n-1\over 10^n},\cdots\right\}$

Given a sequence: $$ \begin{cases} \{x_n\} = \left\{1, \frac{1}{10}, \frac{2}{10},\cdots,\frac{9}{10}, \frac{1}{10^2}, \frac{2}{10^2},\cdots,\frac{99}{10^2}, \cdots, \frac{1}{10^n}, \frac{2}{10^n}\...
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1answer
43 views

Is my intuition correct?

Let $f : X \rightarrow [0,\infty)$ and $A$ , $B\,$ two subsets of $X$ such that $A \cap B \neq \emptyset$. If I have that $ \inf \limits_{X \setminus A} f \,>\, \inf \limits_B f $, does it imply ...
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0answers
24 views

Uniform Law of Large Numbers - questions on supremum, infimum, and weak vs almost sure continuity

I am currently reading up on uniform convergence in probability and I have been trying to familiarize myself with the uniform law of large numbers. As I am not a mathematician or statistician by ...
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2answers
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Positive definite matrix implies the **infimum** of eigenvalues are positive (second version)?

I asked a similar question in here, but actually what I want to ask is more difficult as described below: Suppose $P(x): \mathbb{R} \to \mathbb{R}^{n \times n}$ is always a positive semi-definite ...
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1answer
20 views

Positive definite matrix implies the **infimum** of eigenvalues are positive?

Suppose $P(x): \mathbb{R} \to \mathbb{R}^{n \times n}$ is always a positive definite matrix, does it imply that the infimum (over $\mathbb{R}$) of the minimum eigenvalue of $P(x)$ is always positive?, ...
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1answer
31 views

Proof area of {$(x,y) R^2 | 0 \le y < e^x$ and $0\le x< h$} is $e^h-1$

I'm supposed to prove that the area of {$(x,y) R^2 | 0 \le y < e^x$ and $0\le x< h$} is $e^h-1$ I was going to try to make it a function and calculate it using a Riemanns sum. That led me to ...
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3answers
992 views

Supremum Infimum argument: what did I do wrong?

Suppose $f:[a,b]\to\mathbb R$ be a function such that $|f(x)-f(y)|<\epsilon_0$ for all $x,y\in[a,b]$. Then $M-m\le\epsilon_0$, where $M=\sup\{f(x):x\in[a,b]\}$ and $m=\inf\{f(x):x\in[a,b]\}$. I ...
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4answers
65 views

For a sequence of positive real numbers converging to a limit (which is not equal to 0), show that infimum > 0

I think to do this I either need to prove the infimum is a minumum or a limit but I'm not sure how. I have tried this: By definition of the infimum: " Let A be a subset of the real numbers and b be ...
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2answers
47 views

A sequence with an infimum and no minimum [closed]

If a sequence has an infimum but no minimum, does this mean the infimum is the limit of the sequence? If so how do you prove this?
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1answer
38 views

Sequence of functions on $\mathcal{L}^1([0,1])$ with $\lim_{n\rightarrow \infty}||f_n||_1=0$ but $\sup\{f_n(x): n\in\mathbf{Z}^+\}=\infty$?

Problem Statement. Good evening. As the title suggests, I am having a hard time with the following exercise: Prove that there exists a sequence $f_1,f_2,\ldots$ of functions on $\mathcal{L}^1([...
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1answer
26 views

Find subsequences with specific properties

Let $(a_n)$ be a bounded sequence such that $\inf_{\ell} a_{\ell}<a_n< \sup_{\ell} a_{\ell}$ for each $n=1,2, \dots$ I want to show that there are subsequences $(a_{k_n})$ increasing and $(a_{...
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1answer
42 views

$\sup(A\setminus\{z\}) = \sup(A)$ other direction and generalisation.

Let A be a nonempty set, such that $z\in A$ and $\sup(A) \not \in A$ (a) $\sup(A \setminus \{z\}) = \sup(A)$ (b) generalise this to $\sup(A \setminus \{z_1 z_2 z_3 \dots z_n\})=\sup(A)$ Let ...
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4answers
80 views

Which of the following statements are true for $A=\{t\sin(\frac{1}{t})\ |\ t\in (0,\frac{2}{\pi})\}$?

Let $A=\{t\sin(\frac{1}{t})\ |\ t\in (0,\frac{2}{\pi})\}$. Then $\sup (A)<\frac{2}{\pi}+\frac{1}{n\pi}$ for all $n\ge 1$. $\inf (A)> \frac{-2}{3\pi}-\frac{1}{n\pi}$ for all $n\ge 1$. $\sup (A)...
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2answers
192 views

Solution for Cauchy Problem $u_t-u_{xx} = 0$ belongs to the Gevrey class of order $1/2$

Let $u(x,t)$be the solution for the Cauchy Problem $$u_t-u_{xx} = 0 \mbox{ in $\mathbb{R}\times ]0, \infty[$}$$ $$u(x,0) = u_0(x) \mbox{ in $\mathbb{R}$} $$ where $u_0\in S(\mathbb{R})$(...
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0answers
109 views

Solution for the heat equation that doesn't belong to the Gevrey Class

Show that there exists a solution for the heat operator (in one spatial variabe) that doesn't belong to the Gevrey class of order $s$ for all $s<2$ I already defined the Gevrey class here: $\...
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2answers
139 views

Finding $\sup\limits_{\lambda\ge 0}\{\lambda^ke^{-a\lambda^2/2}\}$

c) Let $a>0$. Find, for each $k=0,1,\cdots$, $$\sup_{\lambda\ge 0}\{\lambda^ke^{-a\lambda^2/2}\}.$$ d) Define, for $x\in\mathbb{R}$, $$v(x) = \int_{\mathbb{R}}e^{ix\lambda-a\lambda^2} ...
1
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1answer
34 views

Sequence $a_n$ such that $\inf a_n < \lim \inf a_n < \lim \sup a_n < \sup a_n$

I'm searching for a sequence with the following property: $$\inf a_n < \lim \inf a_n < \lim \sup a_n < \sup a_n$$ I am looking for just one example, but I am not sure how to find one.
3
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1answer
95 views

$\sup_K |\partial^{\alpha}u|\le C^{|\alpha|+1}\alpha!^s$ then $u$ is analytic for $s\le 1$

A function $u\in C^{\infty}$ belongs to the Gevrey Class of order $s$ if for every compact $K$ of $\Omega$ there is a constant $C$ such that $$\sup_K |\partial^{\alpha}u|\le C^{|\alpha|+1}\alpha!^s$$ ...
0
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1answer
24 views

Equivalence in $\infty$-norm

If $\overline{M}:=\inf\left\{M:\mu(\left\{x:|f(x)|>M\right\})=0\right\}$ and $\overline{a}=\sup\left\{a:\mu(\left\{x:|f(x)|>a\right\})>0\right\}$ I want proves $\overline{a}=\overline{M}$ ...
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2answers
18 views

How to show Sup over bigger set is bigger and Inf over bigger set is lesser?

Let $A,B \in \mathbb{R}^n$ be two subset such that $A \subseteq B$. Also, let $f:\mathbb{R}^n \rightarrow \mathbb{R}$ be a real-valued function. I always use $$\sup_Af(x) \leq \sup_Bf(x)$$ That ...
0
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1answer
20 views

Let $f:X\rightarrow \Bbb{R}\,\cup\,\{+\infty\}$ be a map. Then, for $x_0\in X,\;f(x_0)\geq \sup\limits_{V\in U(x_0)}\inf\limits_{x\in V}f(x)$

Let $f:X\rightarrow \Bbb{R}\,\cup\,\{+\infty\}$ be a map where $X$ is a real normed space. Then, for an arbitrary $x_0\in X,$ I want to prove that the following always holds: \begin{align}f(x_0)\geq ...