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
votes
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 :
$$\|g(u) - g(v) \| \leq \|u-v\| \; \...
-1
votes
2answers
24 views
How to prove that infimum of this set is $0$
How do I prove that the infimum of the set $A=\{x+\frac{1}{n}:x\in(0,1),n\in\mathbb{Z^+}\}$ is $0$?
Clearly $0$ is a lower bound of A since $1/n$ is greater than $0$ for all $n$. How do I show that ...
0
votes
3answers
32 views
I want to know if my proofs about the supremum are correct.
Let $x \in \Bbb R$. Prove that $x = \sup\{q \in \Bbb Q: q \lt x\}$.
Proof 1:
Let $x \in \Bbb R$ and let $S=\{q \in \Bbb Q: q \lt x\}$.
Since $\Bbb Q$ is dense in $\Bbb R$, there exists an $r \in \...
0
votes
1answer
24 views
Supremum over integral of bounded and continuous functions is $+\infty$
Let $X, Y$ be Polish spaces with probability measures $\mu, \nu$, respectively; and let $\gamma$ be a finite measure on the product $X \times Y$. We consider the maximization problem
$$
\sup_{\varphi,\...
0
votes
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
vote
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
votes
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
votes
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
votes
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, ...
1
vote
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:
$$ \sup_{Z \in \mathcal{Q}} E\left[\varphi^{(1)} E\left[...
0
votes
1answer
22 views
Can $\sup\{\| x_1 - x_2 \| : x_1,x_2\in X, \|x_1+x_2\| = 2, \|x_1\| = \|x_2\| = 1\} = 0?$
Let $(X,\|\cdot\|)$ be a normed space
Define
$$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
$$S ||u||_{p}^{2} \leq ||u||_{H^{1}_{0}(\Omega)...
0
votes
1answer
38 views
How do I find the infimum of a function using Calculus III techniques?
Let $f(x, y, z) = \frac{x + y + z}{2} - \sqrt{xyz}$. Find the infimum of the function where the domain is restricted to the first quadrant.
There are techniques in Calculus III involving the hessian ...
1
vote
1answer
54 views
Simple proof that $\sup\{b^t : t \in \mathbb{Q} \text{ & }t≤x\} =\sup\{b^t : t \in \mathbb{Q}\text{ & }t<x\}$
Fix $b>1$. Let $B(x) = \{b^t : t \in \mathbb{Q}\text{ & }t≤x\}$ and let $B'(x) = \{b^t : t \in \mathbb{Q}\text{ & }t<x\}$.
Show that $\sup B(x) = \sup B'(x)$. It is quite easy to show ...
0
votes
0answers
15 views
Suprema, Absolute Value and Expectation Inequality
Problem
Given dataset $S=\{(\mathbf{x}_1,y_1),\cdots, (\mathbf{x}_n,y_n\}$, loss function $\ell(\cdot,\cdot)$ and hypothesis $h\in\mathcal{H}$, define
$$
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
votes
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
votes
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}&...
