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).
3,005 questions
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Does sequential compactness guarantee every convergent subsequence in S converges to a point in S
Let $E,F \subseteq \mathbb{R}^d$ be compact sets. Define
$$ d(E,F) := \inf_{x \in E, y \in F} |x - y|$$
Prove that there exists points $\hat{x} \in E,\hat{y} \in F$ such that
$$ d(E,F) = |\hat{x} - \...
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How to estimate the supremum of absolute value of the $k$-th derivative of $e^{\frac{1}{x^2-1}}$ in $(-1,1)$?
I want to know, is there a function $f$ with simple form, such that $$\sup_{x\in(-1,1)}\left|\partial ^k_xe^{\frac{1}{x^2-1}}\right|<f(k) $$
I calculated the values:
$$
\begin{array}{|c|c|}
\hline
\...
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How to find $\alpha$ where :$\text{sup}(|f|+|g|)\ge \alpha(\text{sup}(|f|) +\text{sup}(|g|))$
$f$ and $g$ two function continu
How to find $\alpha$ where :$$\text{sup}(|f|+|g|)\ge \alpha(\text{sup}(|f|) +\text{sup}(|g|))$$
I know this is true :
$$\text{sup}(|f|+|g|)\le \text{sup}(|f|) +\text{...
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A question about interchanging order of supremum and limit
I encountered some gaps in Amann and Escher’s Analysis I
the hypothesis of theorem3.2 is the function f is n-times continuously differentiable and its domain is a convex perfect subset of $\mathbb{K}(...
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$A = \left\{ \frac{x}{x+1} : x > 0 \right\}$. Prove inf $A$ = $0$.
$A = \left\{ \frac{x}{x+1} : x > 0 \right\}$
The set A is bounded, because $0 < 1 - \frac{1}{x+1} < 1$ for all $ x > 0 $.
My attempt:
inf $A$ = $0$, because:
$ 0 < 1 - \frac{1}{x+1}$ ...
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Does $\sup_{x \in L} |f'(x)|$ always exist if $f$ is differentiable? ("Introduction to Analysis I" by Mitsuo Sugiura.)
I am reading "Introduction to Analysis I" by Mitsuo Sugiura.
Theorem
Let $U$ be an open subset of $\mathbb{R}^n$, and let $f : U \to \mathbb{R}^m$ be a differentiable function on $U$. ...
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Prove that $\mu(B\setminus A_\infty)=0$
Let $(X,\mathcal{E},\mu)$ be a probability space and $\mathcal{A}\subseteq\mathcal{E}$ a family closed with respect to countable unions and intersections that contains the empty set. From the ...
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Prove $B:=\left\{\frac{y}{y-1}:y \in \mathbb{R} \setminus \{1\}\right\}$ is not bounded below nor above.
Given $$B =\left\{\frac{y}{y-1}:y \in \mathbb{R} \setminus \{1\}\right\}$$
prove that $B$ is not bounded below nor above.
I was thinking of using contradiction of the lemma below. Is that correct, or ...
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When does the AM-GM inequality determine the infimum of a set?
Consider the set $S=\{x+x^{-1}:x\in\mathbb{Q},x>0\}$. Using the AM-GM inequality, we may obtain $$x+x^{-1}\geq 2\sqrt{x(x^{-1})}=2.$$ We may also show that $1+(1)^{-1}=2,$ but can we conclude, ...
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Uniform convergence of $f(x)=\sum\limits_{n=1}^{+\infty}\frac{x^n}{n^3+x^2}$
Let's consider $$f(x)=\sum\limits_{n=1}^{+\infty}\frac{x^n}{n^3+x^2}$$
First, I check the convergence of the series $\sum\limits_{n=1}^{+\infty}\frac{x^n}{n^3+x^2}$ to find the domain $D$ of $f$. I ...
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Sufficiency for Sup (C) = Sup(A) + Sup(B)
This came from my last week's midterm exam problem.
This Course is Advanced Calculus/Introduction to Analysis, and the textbook is Apostol's.
My professor asked us to:
Prove if the claim is believed ...
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It is possible to find $A,B \subset R$ such that $\sup A=\sup B$ and $A \cap B = \emptyset$?
It is possible to find $A,B \subset R$ such that $\sup A=\sup B$ and $A \cap B = \emptyset$ and $\sup A \not\in A$ and $\sup B\notin B$?
i think i can consider $S=\mathbb{Q} \cap (0,1)$ and $T=\mathbb{...
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$\nu_t=\text{ess sup}_T\frac{E[\sum^T_{s=t+1}c^s R_s|F_t]}{E[\sum^T_{s=t+1}c^s|F_t]}$ implies $\text{ess sup}_TE[\sum^T_{s=t+1}c^s(R_s-\nu_t)|F_t]=0$?
Let $(R_s)^\infty_{s=1}$ be a bounded process
adapted to a filtration $(\mathscr{F}_s)^\infty_{s=1}$,
let $0 < c < 1$,
and for each integer $t > 0$ let
\begin{equation*}
\nu_t =
\text{ess ...
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Confused by two different answer: supremum in $\mathbb{Q}$
So I was going to check if I could be right in this: $\sqrt{2} \not\in \mathbb{Q}$ hence if a function, in $\mathbb{R}$ has a maximum at that point, studying it over $\mathbb{Q}$ wouldn't lead to the ...
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Find maximum, minimum, supreme and infimum of : $A_n = \{ f'(a) : f(x) = x^n \text{ and } a \in (0,1) \}$
Someone can please help me out with this exercise, I have to find the minimum, maximum, supremum and infimum of:
$A_n = \{ f'(a) : f(x) = x^n \text{ and } a \in (0,1) \}$
I calculated the derivative: $...
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Do there exist any analytical bounds on the Hurwitz Zeta function?
For the general real-valued Hurwitz Zeta Function
$$
\zeta\left(s,a\right),\quad s > 1,\ a > 0.
$$
Do there exist upper and/or lower bounds ?.
Even some ...
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Infimum of a functional in a set of continous functions
The set $A$ is defined as $A=\left\{ f\in\mathcal{C^{1}}\left(\left[0,1 \right],\mathbb{R} \right),f(0)=1 \right\}$, and we define the functional $\phi$ as $\phi(f)=\int_{0}^{1}f(x)dx$. I'm asked ...
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Definition of $\eta_t$ is ambiguous (?)
I'm reading the paper Limit Theorems for Multitype Continuous Time Markov Branching Processes, I. The Case of an Eigenvector Linear Functional (1969) by Athreya, and in the proof of Lemma 3 it says, ...
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Prove $\sup_{P=(t_0,\dots,t_k)}\sum_{j=1}^k |f(t_j)-f(t_{j-1})|=\sup_{n\in\mathbb N}\sum_{j=1}^{k_n} |f(t_j^n)-f(t_{j-1}^n)|$
Let $f:[0,1]\rightarrow \mathbb R$ be right continuous $P_n=(0=t_0^n<t_1^n<\dots<t_{k_n}^n=1)$ a sequence of grids with $\max_{1\leq j\leq k_n} |t_j^n-t_{j-1}^n|\to 0 (n\to \infty)$. Show ...
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Prove that $ d(p,q)= \sup\{|a_i -b_i| : i\in \mathbb N \cup\{0\}\}$ is a metric on Poly$(\mathbb R)$
Consider the collection Poly $(\mathbb R)$ of polynomial functions from $\mathbb R$ to $\mathbb R$.
Each member of Poly $(\mathbb R)$ can be represented as a sum $\sum_{i=0} ^{\infty} a_i x^i$, where ...
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Existence of sequence converging point-wise to sup in chain of uniformly bounded functions
I am quite new to posets, and I am trying to understand what I believe is a well known property (or a claim that is well known to be false in general).
Let $X$ be a set and $\mathcal F_X$ denote the ...
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Distance function to unit fractions — where differentiable?
Define a function $f\colon [0,2] \rightarrow \Bbb R$ by $$f(x) = \inf\left\{\left| x-\frac{1}{n}\right|: n \in \Bbb N\right\}$$ determine the set of points at which $f$ is differentiable.
My solution:
...
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Complex analysis computing the supremum of a set
Consider the set
$$A = \{\vert z\vert: e^{-\frac{1}{z}}= z\},$$
where $z\in \mathbb C$.
I want to show that $\sup A < 1$. We were given as a hint that $e^z = z$ only has solutions for $\vert z \...
user1457233
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What is the inequality connecting $\sup(F_n+G_n+H_n)$ and $\sup(F_n)+\sup(G_n)+\sup(H_n)$?
We have a sequence of functions $K_n$, where
$$K_n=F_n+G_n+H_n$$
So, is $$\sup(K_n)\le \sup(F_n)+\sup(G_n)+\sup(H_n)?$$
$$\forall n\ge 1$$
Also, what will the infimum of the sum be like?
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On a complete metric field, does closed bounded subspaces have extrema (and contain them)?
Saying it otherwise, do such subspaces have the least-upper-bound property, which is useful in many analysis proofs? How can it be proved?
To be clearer:
let $\mathbb{K}$ be a field;
let $d:(x,y)\in \...
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Proof of completeness of $\mathbb{R}$
I was struggling with the proof I found in a book (Analisi Matematica 1, Pagani-Salsa), about completeness of $\mathbb{R}$ that is quite different from every other proof of it I've ever seen. The ...
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Proving $\sup(A) = \sup(X)$ where $A = \{ x \in [a, b] \mid f(x) \leq \gamma \}$ and $X = \{ x \in [a, b] \mid \sup(f([a, x]) \leq \gamma \}$
Suppose $f$ is a continuous function on $[a, b]$ and we are given $\gamma$ between $f(a)$ and $f(b)$. $A$ and $X$ are defined as follows:
$$
A = \{ x \in [a, b] \mid f(x) \leq \gamma \}\\
X = \{ x \in ...
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Topology Mendelson Lemma 5.6 Greatest lowest bound clarification
In Mendelson's book [Introduction to toplogy, 3rd edition] on page 50, Lemma 5.6 is stated as:
Let b be the greatest lower bound of the non-empty set $\mathbf{A}$ of real numbers. Then, for each $\...
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Absurd inequality arising while proving the homogeneous property of integrals
To prove $\int_a^b cf(x) dx=c\int_a^b f(x)dx$ for $c>0$. Refer page 85 of apostol calculus for more details.
Let an integrable function $f(x)$ be approximated by all step-functions $t_1(x)$ and $...
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need explanation about proofing limit Superior and inferior
Let $(x_n)$ be a bounded sequence. For all $n \in \mathbb{N}$ defined $s_n = sup\{x_k | k \ge n\}$ and $t_n = inf\{x_k | k \ge n\}$. Prove that $(s_n)$ and $(t_n)$ are convergent.Then prove if $\lim_{...
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How do I find the supremum and infimum of $\left\{x\in\mathbb{R} \setminus 0: \frac{2x}{3}-\frac{x^2-3}{2x} + 0.5 < \frac{x}{6} \right\}$?
$A=\{x \in \mathbb{R}: \frac{2x}{3}-\frac{x^2-3}{2x} + 0.5 < \frac{x}{6}, x \neq 0\}$
I tried simplifying this inequality and got $x+3<0$, which is $x<-3$. But that means that either $x>0$ ...
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If an implication holds pointwise, can we informally take the supremum and the implication will still hold?
Notation and context. Consider the usual $n-$dimensional real space $\mathbb R^n$ and let $\Omega \subset \mathbb R^n$ stand for an arbitrary non-empty open subset. Moreover, consider function $f_{x,r}...
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Proof assistance: show that $y < F(x)$ is equivalent to $F^{-1}(y) < x$
Exercise:
For a distribution function $F : \mathbb{R} \to [0,1]$, let
$$F^{-1} : [0,1] \to \mathbb{R}, \quad u \mapsto \sup \{ x \in \mathbb{R} \mid F(x) \leq u \}$$
be the generalized inverse of $F$.
...
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Existence of the supremum can't be proven using field and order axioms
I'm reading through "Introduction to Real Analysis" by Robert G. Bartle and Donald R. Sherbert, fourth edition, page 39.
But how does he know that the existence of the supremum can't be ...
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Supremum of a function with power
Is it true that for a function $f(x): \mathbb{R} \to [0,\infty)$ it holds
$$\sup[f(x)^a] \leq [\sup(f(x))]^a $$
for $a \in (-\infty,+\infty)$?
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Infimum of a Set $A = \{ n - \frac{1}{n} : n \in \mathbb{N} \} $
I have had trouble proving that the infimum of the set $A = \{ n - \frac{1}{n} : n \in \mathbb{N} \} $ is zero.
It is easy to prove that 0 is a lower bound. What I am struggling with is proving the ...
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Characterizing the asymptotic properties of $f(k)>\frac{ak^2}{k-1}$
Context: Let $a>0$ be some given constant. Let $f:\{2,3,\text{...}\}\to\mathbb{R}_+$ be some increasing function. Consider the following inequality:
$$\qquad f(k)> a\frac{k^2}{k-1}. \tag{$*$} $$
...
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What's wrong with my proof that $\text{sup}(A)+\text{sup}(B)\le\text{sup}(A+B)$ Spivak Chapter 8, Problem 13
According to Spivak
To prove that $\text{sup}(A)+\text{sup}(B)\le\text{sup}(A+B)$ it suffices to prove that $\text{sup}(A)+\text{sup}(B)\le\text{sup}(A+B)+\epsilon$ for all $\epsilon>0;$ begin by ...
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Sequence approaching a sup of an inf [closed]
Let $f(x,y):X\times Y \to \mathbb{R}$ and consider a sequence $x_k,y_k$ such that $\lim_k f(x_k,y_k)=\sup_{x\in X}\inf_{y\in Y} f(x,y)$.
Is it true that $\lim_k \inf_{y\in Y} f(x_k,y)= \sup_{x\in X}\...
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Infimum/Supremum: How could you have quickly completely this MCQ?
Consider sequence $$B=\left\{\frac{(-1)^n}{\sqrt{2n+1}}:n\in\mathbb{N} \right\}$$ What is the $\sup B$ and $\inf B$?
(A) $\sup B=\frac{1}{\sqrt[3]{2}},\inf B=-1$
(B) $\sup B=\frac{1}{\sqrt[3]{2}},\inf ...
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3
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193
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Why is $\sup A-\epsilon<a$ rather than $\sup A-\epsilon\leq a$ for any $\epsilon>0$?
Why is $\sup A - \epsilon < a$ for some $a \in A$? Why isn't it $\sup A - \epsilon \leq a$?
I'm talking about a specific case where $A$ is an open interval, for example $A =(0,1)$. It feels like if ...
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Given a bounded function $f : B \subseteq R \rightarrow R$, prove that: $sup_{x,y \in B}(f(x) - f(y)) = sup_{x \in B}(f(x)) + sup_{y \in B}(-f(y)).$
Suppose that $f : B \subseteq R \rightarrow R$ is a bounded function.
While trying to prove: $$osc(f,B) = sup_{x \in B}(f(x)) - inf_{y \in B}(f(y)) = sup_{x,y \in B}(|f(x) - f(y)|)$$ I had some ...
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measure theory about completeness of measure
Let ($X,\mathcal A$) be a measurable space, and let $\mu$ be a measure on ($X,\mathcal A$) . The completion of $\mathcal A$ under $\mu$ is the collection of subsets $A$ of $X$ for which there are ...
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Finding infimum and supremum of a set of cyclic sums
I am trying to find the supremum and infimum of the following set:
$$
\left\{ \frac{a_1}{a_1 + a_2 + a_3} + \frac{a_2}{a_2 + a_3 + a_1} + \cdots + \frac{a_{n-2}}{a_{n-2} + a_{n-1} + a_n} + \frac{a_{n-...
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Does the existence of an irrational supremum of a sequence of rationals numbers necessarily imply that it is convergent? [closed]
Got this from an exercise from Axler's MIRA supplementary text, or at least i hope to have made an equivalent statement. The original question goes as follows:
Prove or give a counterexample: Suppose ...
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40
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Interpretation of $\max \sup$
I am reading the regret analysis proof of LinUCB given in Lattimore's Bandit Algorithms. He makes the following assumption:
$$ \max\limits_{t\in[n]}\sup\limits_{a,b\in\mathcal{A}_t} \langle\theta^* , ...
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2
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Prove that $\sup(A\cup B)=\max\{\sup(A),\sup(B)\}$
I want to prove $\sup(A\cup B)=\max\{\sup(A),\sup(B)\}$, where $A,B\subset \mathbb R$ are non-empty and bounded sets from above. I have reviewed similar questions and answers, but I intended to ...
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How does Axler know he has found the infimum?
I am reading the following example from Measure, Integration & Real Analysis by Sheldon Axler about the outer measure:
Suppose $\displaystyle A=\{ a_1,a_2,...,a_n \}$ is a finite set of
real ...
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Prove that $c \sup A = \sup(cA)$ for $c>0$.
I'm new to real analysis and trying to prove $\sup(cA)=c\sup(A)$ for $c>0$. Using this definition of least upper bound:
$s=\sup A$, where $s\in \mathbb R$ and $A\subseteq \mathbb R$ if
$\forall ...
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P(sup_t $\vert X^{t}_n - X^{t}\vert$) vs sup_t P($\vert X^{t}_n - X^{t}\vert$) in probability theory
I have a question regarding probability theory which has to do with the notion of uniform convergence in probability. One considers a sequence of random variables $X_n^{t}$ parametrized by a parameter ...
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