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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).

4
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0answers
39 views

Showing that $\inf \{\|u-g(u)\| : u \in C \} = 0.$

Exercise : Let $X$ be a Banach space and $C \subseteq X$ be closed, convex and bounded. Moreover, let $g:C \to C$ be a non-expansive operator, meaning that : $$\|g(u) - g(v) \| \leq \|u-v\| \; \...
-1
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2answers
24 views

How to prove that infimum of this set is $0$

How do I prove that the infimum of the set $A=\{x+\frac{1}{n}:x\in(0,1),n\in\mathbb{Z^+}\}$ is $0$? Clearly $0$ is a lower bound of A since $1/n$ is greater than $0$ for all $n$. How do I show that ...
0
votes
3answers
32 views

I want to know if my proofs about the supremum are correct.

Let $x \in \Bbb R$. Prove that $x = \sup\{q \in \Bbb Q: q \lt x\}$. Proof 1: Let $x \in \Bbb R$ and let $S=\{q \in \Bbb Q: q \lt x\}$. Since $\Bbb Q$ is dense in $\Bbb R$, there exists an $r \in \...
0
votes
1answer
24 views

Supremum over integral of bounded and continuous functions is $+\infty$

Let $X, Y$ be Polish spaces with probability measures $\mu, \nu$, respectively; and let $\gamma$ be a finite measure on the product $X \times Y$. We consider the maximization problem $$ \sup_{\varphi,\...
0
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0answers
6 views

Breaking down $\text{dom}(f)$ to analyze supremum

I am trying to get to the convex conjugate function of the $\ell_1$ norm of a vector $\mathbf{x}$ without using dual norms of indicator functions. My idea so far was: \begin{align*} f^{*}(\mathbf{y})...
1
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0answers
55 views

Why is it necessary to show subsequence convergence in the extreme value theorem?

I’m probably making this more complicated than it needs to be, but I’m trying to figure out why it is necessary to prove the convergence of a subsequence in proving the extreme value theorem (EVT). I ...
4
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0answers
43 views

supremum of additive functions is additive

I need some help for one equality in the following proposition. It was a hint to conclude that $\sup\{f(\cdot):f\in\Phi\}$ is additive. I highlighted it blue. Ultimately I am interested in proving the ...
2
votes
2answers
37 views

Can't finish this proof about supremums :(

Let $S$ be a nonempty bounded subset of $\Bbb R$ and let $k \in \Bbb R$. Define $kS = \{ks:s \in S\}$. Prove: If $k \ge 0$, then $sup(kS) = k \cdot supS$. Okay, here's my attempt at the proof: ...
1
vote
0answers
20 views

constructing a saddle function/inf-sup variational statement

I'm trying to make sense of a variational problem under certain conditions. The problem goes as follows: Consider the scalar valued function, $u(\textbf{A},\textbf{b})$, where $\textbf{A}$ is a ...
2
votes
1answer
54 views

Question about norms and seminorms

Let $(\mathsf{X},\mathcal{X})$ be a measurable space, $\mathcal{F}_b(X)$ be the space of bounded measurable functions, and define the supremum norm as $\Vert f \Vert_{\infty} = \sup\{ f(x) : x \in \...
0
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0answers
21 views

Supremum of a set involving the distance function

Let $(X,\rho)$ be a metric space (nonempty, with infinite cardinality). For a nonempty closed set $F \subset X$, define the distance function, $d(x,F)$, as, $$d(x,F) = \inf\limits_{z \in F}\rho (x,z)$$...
0
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0answers
12 views

$\lim_{c\rightarrow c^*,a\rightarrow 1}\sup_{x\geq 0}|exp(-\int_0^{x/a}(1+ct)_+^{-1})dt-exp(-\int_0^{x}(1+c^*t)_+^{-1})dt| $=0?

I am trying to read the proof of Theorem 1 in Statistical Inference using Extreme Order Statistics by James Pickands III (Link:https://projecteuclid.org/euclid.aos/1176343003) To make story short, ...
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0answers
13 views

Supremum of expectation equals expectation of supremum?

Even though there were some questions related to this one, I couldn't find a proper answer. I am wondering if the following inequality holds: $$ \sup_{Z \in \mathcal{Q}} E\left[\varphi^{(1)} E\left[...
0
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1answer
22 views

Can $\sup\{\| x_1 - x_2 \| : x_1,x_2\in X, \|x_1+x_2\| = 2, \|x_1\| = \|x_2\| = 1\} = 0?$

Let $(X,\|\cdot\|)$ be a normed space Define $$r=\sup\{\| x_1 - x_2 \| : x_1,x_2\in X, \|x_1+x_2\| = 2, \|x_1\| = \|x_2\| = 1\}.$$ Question: Is $r=0?$ Clearly triangle inequality implies that $r\...
1
vote
2answers
80 views

To prove $\sup B = \inf A$

Let $A$ be bounded below and $B = \{b \in \mathbb{R} : b$ is a lower bound for $A\}$. My Work Now to understand I assumed some values, like set $A = (1,4)$ and $\inf A=1$ and so set $B = (-\infty ...
0
votes
1answer
30 views

Supremum vs. Infimum

I learn best with visuals, so I am struggling to understand why the Supremum is called the "Least Upper Bound" and not the "Greatest Upper Bound", like the Infimum is called the "Greatest Lower Bound"....
0
votes
1answer
17 views

Is this an equivalent way of writing supremum

Let $\{A_n\}$ be a sequence of events I have an indicator r.v. $I_n$ where $I_n=1$ if an event $A_n$ occurs and $0$ otherwise Let $X_n = \sum_{k=1}^n I_k$ Is it the same thing to say this $\sup_{n\...
5
votes
0answers
53 views

Supremum of arc lengths of graphs of power towers

Consider the set of all functions of one variable $x\in[0,1]$ that can be constructed from any number of instances of that variable using parentheses and exponentiation only: $$x,\;x^x,\,x^{x^x},\;\...
0
votes
1answer
23 views

How to obtain the boundedness of $\{a_{n}\}_{n}$

Let $\{a_{n}\}_{n\in\mathbb{N}}$ be a sequence such that $\forall n \in \mathbb{N}, a_{n} > 0$. Let also $k_{1}, k_{2} \in \mathbb{R}$ such that $k_{1} > 0$ and $k_{2}\neq 0$. Assume $\sup\...
1
vote
2answers
18 views

question on the meaning of supremum

I know that supremum means least upper bound. If I have a sequence of events, $\{A_n\}_{n=1}^\infty$ then $$\limsup_{n\rightarrow \infty} A_n = \lim_{n\rightarrow \infty} \sup_{j\geq n} A_j$$ I'm ...
1
vote
0answers
39 views

Prove inequality for infimum of expected values

I'm trying to prove the following inequality, but don't have any idea on where to begin. $$\inf_{k=0,1,2,\cdots} \frac{\mathbb{E}[|X|^k]}{\delta^k} \le \inf_{\lambda> 0} \frac{\mathbb{E}[e^{\...
0
votes
1answer
44 views

Prove every $S\subseteq R$ with a LB has a GLB.

Here is the problem in its entirety: Prove that every non-empty $S\subseteq \mathbb{R}$ that has a lower bound has a greatest lower bound. For every set $S$ it will be helpful to think about the ...
-2
votes
0answers
18 views

supremum of an expectation

I'm really struggling to understand why \begin{align} \sup_{n\in N} E[\left\lvert x_n \right\rvert 1_{\left\lvert x_n \right\rvert \ge k}] \end{align} is \begin{align} \le1 \end{align} Can someone ...
0
votes
0answers
16 views

Supremum and infimum of a stochastic process

Suppose $\{X_t\}_{t\geq 0}$ is a real-valued stochastic process satisfying $E[X^2_t]\rightarrow\infty$ as $t\rightarrow\infty$. Does it hold that $\sup |X_t|\rightarrow \infty$? Intuitively I think ...
0
votes
1answer
27 views

Jensen's inequality with supremum

Problem In a paper I am reading now, the author claims that by Jensen's inequality, they have $$ \frac { 1 } { \lambda } \log \exp \left( \lambda \cdot \mathbb { E } _ { \epsilon } \sup _ { h \in \...
3
votes
1answer
40 views

Supremum of $M = \{ \left \lfloor{\alpha n} \right \rfloor\frac{1}{n}: n \in \mathbb{N}_{>0}\}$

Lets assume the set $M = \{ \left \lfloor{\alpha n} \right \rfloor\frac{1}{n}: n \in \mathbb{N}_{>0}\} $ with $\alpha \in \mathbb{R}, \alpha > 0$. How can I systematically find the supremum of ...
5
votes
0answers
38 views

No rational supremum proof

I'm currently in the beginning of a journey of self-studying real analysis, and I tried to prove the following statement: Prove that the following set does not have a rational supremum: $$ E=\{x \in \...
1
vote
1answer
24 views

Approximating a sum by the minimum of a subset times its size

I cannot believe the following infimum unequals zero. Can anybody find a way to construct a sequence of subsets for which the expression in the infimum tends to zero? $$\inf\left\{\frac{\sup_{B\...
0
votes
1answer
20 views

Difference of the sup

Let $f, g:A\to \mathbb R$ two functions. Is it true that $$\sup_{x\in A}|f(x)|-\sup_{x\in A}|g(x)|\leq \sup_{x\in A}|f(x)-g(x)|?$$
1
vote
2answers
34 views

Is the sup of continuous functions still continuous?

Suppose I have a family of functions $\{f_t, t\in [0, T]\}$, where $f_t:A\to\mathbb R$, with $A$ a generic set (not necessarily contained in $\mathbb R$). Suppose that, for all $t\in [0, T]$, $f_t$ is ...
0
votes
2answers
23 views

Show that $I=(−\infty,\sup I]$ as $I$ is bounded above but not bounded below

Let $I$ be a non-empty interval. Suppose $I$ is not bounded below, I is bounded above, and $\sup I ∈ I$. Show that $I=(−\infty,c]$, where $c=\sup I$. My attempt:($\Longrightarrow$) Since $I$ is not ...
0
votes
1answer
19 views

Norm Inequality for 1 Dimensional Sobolev Space

Let $\Omega \subset \mathbb{R}$ be an unbounded domain and $u \in H_{0}^{1}(\Omega)$. By Sobolev Embedding Theorem for 1 dimensional space, we can obtain $$S ||u||_{p}^{2} \leq ||u||_{H^{1}_{0}(\Omega)...
0
votes
1answer
38 views

How do I find the infimum of a function using Calculus III techniques?

Let $f(x, y, z) = \frac{x + y + z}{2} - \sqrt{xyz}$. Find the infimum of the function where the domain is restricted to the first quadrant. There are techniques in Calculus III involving the hessian ...
1
vote
1answer
54 views

Simple proof that $\sup\{b^t : t \in \mathbb{Q} \text{ & }t≤x\} =\sup\{b^t : t \in \mathbb{Q}\text{ & }t<x\}$

Fix $b>1$. Let $B(x) = \{b^t : t \in \mathbb{Q}\text{ & }t≤x\}$ and let $B'(x) = \{b^t : t \in \mathbb{Q}\text{ & }t<x\}$. Show that $\sup B(x) = \sup B'(x)$. It is quite easy to show ...
0
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0answers
15 views

Suprema, Absolute Value and Expectation Inequality

Problem Given dataset $S=\{(\mathbf{x}_1,y_1),\cdots, (\mathbf{x}_n,y_n\}$, loss function $\ell(\cdot,\cdot)$ and hypothesis $h\in\mathcal{H}$, define $$ u=\frac{1}{n}\sum_{i=1}^n\ell(h(\mathbf{x}_i,...
2
votes
1answer
57 views

Bounding in Glivenko-Cantelli theorem

Problem Let $X_1, X_2, \cdots, X_n$ be iid random variables. The cdf and empirical cdf are $F(t)=P[X\leq t]$ and $\hat{F}_n(t)=\frac{1}{n} \sum_{i=1}^n 1(X_i\leq t)$. The Glivenko-Cantelli theorem ...
0
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0answers
31 views

Can someone guide if I'm proceeding correctly with solving the below problem?

$A\in R^{n\times d}$ : consider it to be full rank. $x \in R^{d}$ $A = U \Sigma V^{T}$ : SVD decomposition of A: I'm suppose to find the value of below terms $\sup_x \frac{|| \Sigma V^{T}x||_{\infty}...
0
votes
1answer
33 views

Does $P(\liminf_{n \to \infty}\{|X_{n}|\leq \epsilon\})=1\iff \exists N \in \mathbb N, |X_{n}|\leq \epsilon, \forall n \geq N, P-$a.s.

Background to my question: Given that $(X_{n})_{n}$ are random variables on $(\Omega, \mathcal{F}, P)$ and for $\epsilon > 0$: $\sum_{n \in \mathbb N}P(|X_{n}|>\epsilon)<\infty$ It follows ...
0
votes
1answer
21 views

Infimum of positive periodic function is positive

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a positive, smooth, periodic function with period $T$. I need to show that the infimum of $f$ is positive. My idea is that on each interval of a period $f$...
0
votes
1answer
29 views

How to determine the supremum of a set with respect of all vectors of norm 1?

I'm having some troubles regarding the definitions that use the supremum of a set for all vectors with a specific norm. At the moment the case in question is the operator norm, defined as $\|T\|_{X'} ...
2
votes
1answer
29 views

$\limsup_{n \to \infty}\{\frac{X_{n}}{\log{(n)}}\leq\epsilon\}\subseteq \{\liminf_{n \to \infty}\frac{X_{n}}{\log{(n)}}\leq\epsilon\}$

I recently saw an assertion made that $$\bigcap_{m \in \mathbb N} \bigcup_{n \geq m}\left\{\frac{X_{n}}{\log{(n)}}\leq\epsilon\right\} \subseteq\left\{\liminf_{n \to \infty}\frac{X_{n}}{\log{(n)}}\...
0
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0answers
37 views

Question on $\liminf A_{n}$ and $\limsup A_{n}$

Say I have the following event $\{\limsup_{n \to \infty} |X_{n}|> \epsilon\}$, and that $P(\{\limsup_{n \to \infty} |X_{n}|> \epsilon\})$, I also realize that $\{\limsup_{n \to \infty} |X_{n}|&...
1
vote
0answers
26 views

Limit (involving $\sup$) - proof verification

Suppose $\phi:\mathbb{R}^n\to \mathbb{R}$ is a $C^1$ function and fix $x \in \mathbb{R}^n$. For every $t \in (0,+\infty)$ let $y_t:[0,+\infty)\to \mathbb{R}^n$ be a continuous function such that $y_t(...
1
vote
2answers
47 views

Proving the supremum of a set

Let A be a nonempty subset of $\mathbb R$ and let m be an upper bound of A. Prove that $m = supA$ iff $ \forall n \in \mathbb N, \exists a \in A, m < a + \frac 1n$. I don't really have any idea ...
0
votes
1answer
34 views

Showing infimum of a set is smaller or equal to the infimum of a different set

Let $R= [a,b] \times [c,d]$ be an arbitrary rectangle. Define $S_1 =\{f(x,y): (x,y)\in R\} $ and $S_2=\{f(x,y):a\leq x \leq b\}$ Claim: For every $c\leq y \leq d$ we want to show that $\inf(S_1) \...
0
votes
1answer
22 views

Supremum of product of two functions

If I am given two continuous functions $ f, g : [a,b] \rightarrow \mathbb{R}$ with $g(x)\geq 0$, then is it true that $\sup(fg) \leq \sup(\sup(f) g)$? The stronger conjecture that $\sup(fg) \leq \...
1
vote
2answers
67 views

Infimum of $n$-th root of $n$

Let $A = \left \{ \sqrt[n]{n} \mid n \in \mathbb{N}\right \}$. I need to find and prove the infimum of $A$. Because $n \in \mathbb{N},$ we can know for sure that $\sqrt[n]{n} \geq 1$. (Does that ...
1
vote
0answers
38 views

Properties of a supremum of a parametrized set II

Please I need help with this question: $E(x)= \left \{ \left(1+ \frac{x}{n} \right)^n : n \in \mathbb{N} \right \}$. Let $a(x) = \sup E(x)$ (least upper bound). ($1$) Prove that $a(x) < a(y)$ if $...
0
votes
0answers
53 views

Properties of a supremum of a parametrized set

Please I need help solving this question: $E(x)= \left \{ \left(1+ \frac{x}{n} \right)^n : n \in \mathbb{N} \right \}$. Let $a(x) = \sup E(x)$ (least upper bound). (1)Prove that $a(x) < a(y)$ if $...
0
votes
0answers
8 views

Defining different cases of Epsilon while finding the infimum of a group

Let $ A = \left \{ \frac{n-m}{n+m}\mid n,m \in \mathbb{N}, m<n \right \} $ I need to prove the the infimum of $ A $ is zero. So I get: $ \frac{n-m}{n+m} = 1 - \frac{2m}{n+m}\geq 1-\frac{2}{n+1}&...