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).
2,798
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Let $f: [a,b] \to \Bbb R$ be continuous. Then $f$ is bounded. Prove using sequence and subsequences. [closed]
Let $f: [a,b] \to \mathbb{R}$ be continuous. Then
$f$ is bounded.
Let $M:= \sup\{f(x): x \in [a, b]\}$ and $m := \inf\{f(x): x \in [a, b]\}$. Then there exist points $c$ and $d$ in $[a, b]$ such that ...
0
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Is given statement indeed true?
I saw saw the statement I am trying to prove in "a proof of lub property" in a Mathematics Stack Exchange post. The statement I am trying to prove is :
$∃\ b ∈ \mathbb {R},(b < a\
\text { ...
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2
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Proof of $\inf\left \{ \frac{\mathrm{d} (n^2)}{\mathrm{d} (n)} \; \bigg| \; n \in \mathbb{N} \right \}=0$, where $d(n)$ is the sum of digits of $n$
So I wanted to find the infimum of the set described in the title, and I'm pretty confident on what the subsequence should be to ensure a 0 infimum.
But a proof of this eludes me. I tried some funky ...
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1
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Suppose function $f(x)$ is continuous on $[a,b]$ then the function $\sup_{a\le{t}\le{x}}f(t)$ is also continuous.
In this problem, I need to prove $\sup_{{a}\le{t}\le{x}}f(t)$ is continuous, and I denote this function by $M(x)$. It is obvious that the function $M(x)$ is a non-decreasing function，so I can conclude ...
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Show that there exists $0 \neq v_0 \in H$ with $v_0 \perp V$ such that the supremum is attained.
Let $H$ be a separable Hilbert space and $V$ be an $m$-dimensional proper subspace of $H.$ Let $T \in \mathcal L (H)$ be a compact operator. Show that there exists $0 \neq v_0 \in H$ with $v_0 \perp V$...
- 2,056
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43
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Infimum of the set $\{(1/m)^{1/n}+(1/n)^{1/m}\mid m,n\in\Bbb N\}$
I have to find an infimum of the set $A:=\left\{\frac{1}{\sqrt[m]{n}}+\frac{1}{\sqrt[n]{m}}: m,n\in\mathbb{Z}_+\right\}$. I think that it is $1$. It is easy to find a sequence of numbers from $A$ that ...
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Inf and sup on $A=\left\{(x_1,x_2,\dots,x_{10}) \in \mathbb R^{10}: \sum_{1 \le i<j \le 10} x_ix_j=45, x_1\ge0, x_2\ge 0,\dots, x_{10} \ge 0 \right\}$
Let $$A=\left\{ (x_1,x_2, \dots, x_{10}) \in \mathbb R^{10}: \sum_{1 \le i < j \le 10} x_ix_j=45, x_1 \ge 0, x_2 \ge 0, \dots, x_{10} \ge 0 \right\}$$ and let $f(x_1,x_2, \dots, x_{10})=x_1+x_2+\...
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2
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Find an upper bound for $\int_{0}^{1}\cos(x)\frac{x f'(x) - f(x) + f(0)}{x^2}dx$
Let $f : [0, 1] \to \mathbb{R}$ such that $f$, $f'$ and $f''$ are continuous on $[0, 1]$ We want an upper bound for
$$\int_{0}^{1}\cos(x)\frac{x f'(x) - f(x) + f(0)}{x^2}dx$$
which is strictly bounded ...
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finding infimum of the ratio of largest to smallest distance of six points in $\mathbb R^2$
How can we find the infimum of the following set
$$\left\{\frac{\max_{i\neq j}d(A_i,A_j)}{\min_{i\neq j}d(A_i,A_j)}:A_1,A_2,\ldots,A_6 \in \mathbb R^2, A_i\neq A_j (\forall i\neq j)\right\}.$$
This ...
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2
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Find infimum and supremum of the function $f$ defined by the formula $f(x, y, z) = 2x+ 2y-3z$ on the set $A$
Let $A = \{(x, y, z) \in \mathbb R^3 : x^2 + y^2 + z^2 = 2, xy + yz +zx +1 = 0\}$. Find infimum and supremum of the function $f$defined by the formula $f(x, y, z) = 2x+ 2y-3z$ on the set $A$.
I want ...
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Definition od essential supremum for martingales with many measures
Consider the measurable space $(\Omega,\mathbb{P},\mathcal{F})$ and a filtration $\mathbb{F}=\left\{\mathcal{F}_t\mid\,t\in[0,T]\right\}$. Let $\mathcal{Q}$ be a family of equivalent probability ...
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Let $S=\{\frac{x+1}{x^2+1}:x\in \mathbb{Q},x>0\}$ Prove that $\sup S=\frac{1+\sqrt{2}}{2}$
Let $S=\{\frac{x+1}{x^2+1}:x\in \mathbb{Q},x>0\}$. Only use the knowledge about supremum and infimum. Prove that $\sup S=\frac{1+\sqrt{2}}{2}$.
I know the basic method the find the supremum and ...
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Swapping an inf sup and a limit involving a probability
Let $\mathbb{P}$ be a probability measure, and denote by $f$ an arbitrary function that takes as inputs a random variable $X$ and a parameter $\theta \in \mathbb{R}^d$. I know that for every integer $...
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1
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Existence of supremum implies existence of infimum
I'm reading A Book of Set Theory by Charles C. Pinter. I don't know how to solve exercise 4.3.11.
Let $A$ be a partially ordered class. Prove the following:
a) If every subclass of $A$ has a $\sup$ ...
3
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1
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If $0<a<1$, then $\inf\{a^n:n\in\mathbb{Z}_+\}=0$.
Suppose $0<a<1$. I’m trying to prove $\inf\{a^n:n\in\mathbb{Z}_+\}=0.$ There’s just one step in my proof I would like confirmation on.
First we note $0$ is clearly a lower bound for the set $\{a^...
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2
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$F$ is lower semicontinuous $\iff F(x)=\sup_{r>0}\inf_{y\in B(x,r)}F(y)$ for all $x\in X$
Let (X,d) be a metric space. $F: X\to \overline{\mathbb{R}}$.
The definitions that I have to use are:
(1) $F$ is sequentially lower semicontinuous if for all sequences $(x_n)_n \subseteq X$ s.t $ x_n\...
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1
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Sufficient condition that the Infimum of an uncountable family of measurable functions is measurable
Let $(\Omega, \mathcal{F})$ be a measurable space and $(f_r)_{r \in \mathbb{R}}$ be a family of $\mathcal{F}-\mathcal{B}(\mathbb{R})$-measurable functions $f_r : \Omega \to \mathbb{R}$. In general, ...
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(X,d) be a metric space. $F: X\to \overline{\mathbb{R}}$. $F$ is sequentially lower semicontinuous iff $F$ is lower semicontinuous
Let (X,d) be a metric space. $F: X\to \overline{\mathbb{R}}$. $F$ is sequentially lower semicontinuous iff $F$ is lower semicontinuous
The definitions that I have to use are:
(1) $F$ is sequentially ...
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2
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Prove that $\sup Z=\frac {\sup X}{\inf Y}$
Let $X$ and $Y$ be two nonempty bounded sets of positive real numbers.
Define $Z=\{\frac{x}{y}:\space x\in X,y\in Y\}$.
If $\inf Y>0$, prove that $\sup Z=\frac {\sup X}{\inf Y}$
This is one of the ...
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Uncountable ordered sets and monotonic functions: defining a sequence with a particular inferred restriction via induction
For this question, let us assume $S$ is an uncountable nonempty ordered set such that every its nonempty subset has an infinum and a supremum. And let us assume that $f \colon S \to S$ is a ...
1
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1
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103
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Find infimum and supremum of the function $f$ defined by the formula $f(x, y, z) = x^2 + y^2 + z^2$ on the set $A$
Let $A = \{(x, y, z) \in \mathbb R^3 : 5x^2 + 5y^2 - z^2 = 0, x + 2y +3z = 20\}$. Find infimum and supremum of the function $f$
defined by the formula $f(x, y, z) = x^2 + y^2 + z^2$ on the set $A$.
...
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1
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Supremum of limit inferior
Define the limit inferior of a sequence of sets as $$\liminf A_n = \bigcup_{N \geq 0} \bigcap_{n \geq N} A_n , $$ and the limit inferior of a sequence as $$\liminf X_n = \sup_{N \geq 0} \inf_{n \geq N}...
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1
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48
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Prove or disprove infimum of B equals to supremum of A
Prove of disprove this statement:
Let A be a subset of $\mathbb{R}$ which is nonempty and bounded above, and let
$B=\{b\in \mathbb{R} : b\space is\space an\space upper\space bound\space for\space A\}$
...
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Supremum of certain integrable functions [duplicate]
I am currently facing problem to solve the following question. Please help!
Let $F = \{f: [1, 3] \rightarrow [-1, 1] \mid f \text{ is continuous and } \int_{1}^{3}f(x)dx = 0\}.$
Find $\sup_{f \in F}\...
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The greatest lower bound of $|\cos(x)+\sin(x)|$
I was going through a problem when I stumbled across the need to find the infimum of :
$$|\cos(x)+\sin(x)|$$
where $x\in \mathbb R$.
It is easy to get a lower bound, for we must have $|\cos(x)+\sin(x)...
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1
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Final part of the proof of $f$ is convex if and only if $f =\sup\{\varphi:\mathbb{R}^M\to \mathbb{R}: \varphi \le f, \varphi \text { affine} \}$
Let $f : \mathbb{R}^M\to \mathbb{R} $
$f$ is convex if and only if $f =\sup\{\varphi:\mathbb{R}^M\to \mathbb{R}: \varphi \le f, \varphi \text { affine} \}$
To prove ($\implies$) I have to follow the ...
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Suppose that you have uncountably many random variables satisfying $P(X_{\alpha} > x) < \eta$, then is $P(\sup X_{\alpha} > x) < \eta$ true?
Let $A$ be a set that is possibly uncountable. Suppose that for each $\alpha \in A$, we associate to it a random variable $X_{\alpha}: \Omega \to \mathbb{R}$. Suppose there exists $x, \eta \in \mathbb{...
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Understanding Essential sup norm with simple functions
Suppose we have a fixed measure space $(X,M,\mu)$
$||f||_\infty=\inf\{a\geq 0: \mu\big(\{x:|f(x)|> a\}\big)=0\}$
I been looking at this definition for quite a while, but I don't really understand ...
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Finding the Characteristic Function for the Supremum of a OU Process
I'm currently writing my BSc thesis on Lookback Option pricing under Ornstein Uhlenbeck Processes. I'm using the COS method for this. For this I need to find the fourier transform of the pdf (...
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Converse order relations flip completeness property
To me it seems trivially true that if $A$ is a set ordered by relation $C$ denoted $<_C$ and $A$ has the greatest lower bound property (or least upper bound property), then $A$ ordered by the ...
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If $f:[a,b]\rightarrow \mathbb{R}$ is Riemann integrable does $\lim_{p\rightarrow\infty}(\int_a^b|f(x)|^p)^{\frac{1}{p}}=\sup_{x\in[a,b]}|f(x)|$?
Several posts on this site ask for a proof of the statement
If $f:[a,b]\rightarrow\mathbb{R}$ is continuous, $\lim_{p\rightarrow\infty}(\int_a^b|f(x)|^p)^{\frac{1}{p}}=\sup_{x\in[a,b]}|f(x)|$.
Need $...
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commuting expectation and supremum
Assuming that $u$ is a random variable that follows a known distribution and takes values from $U$, and that $v$ is a probability measure (which may not be relevant), we consider a real-valued ...
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why this submultiplicative function is everywhere finite if is bounded in the neighborhood of one
Let $f$ be a positive function finite everywhere defined in $(0, \infty)$ and
$$M_f(s) = \sup_{x\in (0, \infty)} \frac{f(sx)}{f(x)} $$
Is easy to check that $M_f(x_1x_2)\leq M_f(x_1)M_f(x_2)$ and $\...
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1
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Clarifications needed in an exercise about semilattice and abelian monoids in Arbib and Manes' text
The following question is taken from Arrows, Structures and Functors the categorical imperative by Arbib and Manes
Exercise: A $\textbf{semilattice}$ is a poset in which every finite subset has a ...
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0
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how to prove using definition of supremum [closed]
Given sequences of positive
real numbers $(\alpha_N)_{N=1}^\infty$ and $(\beta_N)_{N=1}^\infty,$ the notation $\alpha_N \lesssim \beta_N$ means that $\sup_N \alpha_N / \beta_N < \infty.$ Likewise, ...
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The proof of additivity of total variation of complex measure by Axler.
I'm reading the proof of "total variation function is positive measure" on p.261 in Measure, Integration and Real Analysis by Axler.
(The book pdf: https://measure.axler.net/MIRA.pdf)
Let $\...
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1
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What is a direct method for finding the supremum of the following inductively defined sequence?
The sequence is $z_1=1, z_{n+1}=\sqrt{2z_n}$. If $\epsilon>0$ is given, we must find $m\in\mathbb{N}$ such that $2-\epsilon<2^{1-2^{1-m}}$. But I'm not sure how to find a suitable $m$. I've ...
3
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1
answer
46
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Understanding supremum of ordinals. Differences between limit ordinal and successor ordinal.
I apologize for this naive question, but I'm new to set theory and I'm having a hard time figuring out properties of supremum of ordinals.
From here on out I will refer to $\alpha$ as a nonzero ...
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1
answer
57
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Operator norm for $\|x\|=1$ and $\|x\|<1$ [closed]
let $T:X \to Y$ be a bounded linear map. $X$ and $Y$ are two normed vector space. Each norm is denoted by $\| \cdot \|_X$ and $\| \cdot \|_Y$
define its operator norm by $$\|T\|_\text{op}=\sup_{x \ne ...
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1
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47
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Counter example of a given two sets . [closed]
let $f:(a, b) \rightarrow \Bbb R$ be a differentiable map, and define two sets, $A=\left\{f^{\prime}(x): x \in(a, b)\right\}$ and $B=\left\{\frac{f(y)-f(x)}{y-x}: x \neq y \in(a, b)\right\}$
I know ...
- 474
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0
answers
21
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Relation between sum of sup inf and sup inf of a sum
Given two functions $f\,g : X \times Y \mapsto \mathbb{R}$, I am wondering if there is a general inequality between
$$ \sup_x \inf_y f\left(x,y\right) + \sup_x \inf_y g\left(x,y\right) $$ and $$ \...
- 1
6
votes
1
answer
86
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Prove that $f(s) = \sup\{x \in [a,b] : f(x) > x\}$, where $f$ is increasing
Given $f: [a,b] \rightarrow [a,b]$ increasing, where $f(a) > a$ and $A = \{x \in [a,b] : f(x) > x\}. \ $ If $s = \sup(A), $ prove that $f(s) = s$.
In order to show that $f(s) = s$, I was trying ...
- 295
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0
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48
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First decomposition theorem of fuzzy set
I read if standard union in fuzzy set have definition:
Union of two fuzzy sets $\tilde{A}$ and $\tilde{B}$ in universe $X$ denoted $\tilde{A}\cup\tilde{B}$ is fuzzy set in universe $X$ with membership ...
1
vote
1
answer
28
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Question about notation in the proof that if the lim(s_n) is defined, the liminf(s_n) = lim(s_n).
I am reading Elementary Analysis by Kenneth Ross and am really struggling, especially with the choice of notation and why certain things are included in the proofs.
In chapter ten he is proving that ...
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1
answer
45
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Sup Inf Inequality
I try to solve the following problem:
Let $\varphi: D \to \mathbb{R}$ with $-\varphi(x) = \varphi(-x)$. Then $\underset{x \in D}{sup} \;(\varphi(x) - \varphi(\tilde{x}))\leq \underset{x\in D}{sup} \...
1
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0
answers
58
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If $A, B \subseteq \mathbb{R}$ are bounded intervals with $A \cap B \neq \varnothing$, then $\sup(A \cap B) = \min\{\sup A, \sup B\}$
I am looking for hint on the question:
If $A, B \subseteq \mathbb{R}$ are bounded intervals with $A \cap B \neq \varnothing$, then $\sup(A \cap B) = \min\{\sup A, \sup B\}$.
So far, I have shown ...
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1
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286
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Are inf and sup continuous functionals in general?
Let $X$ be any topological space and $\bar{\mathbb{R}} = [-\infty, \infty]$ with the standard topology. Is it true, in general, that the functionals $$\inf: C(X,\bar{\mathbb{R}}) \to \bar{\mathbb{R}}, ...
- 1,128
1
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0
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26
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The distribution function of a running supremum of a diffusion process in a particular limit.
Let $x>0$, $\theta \le 1$, $\mu_0 \in {\mathbb R}$ and $\nu:= -{\bar \mu_0}/(1-\theta)$ and ${\bar \mu_0} := \mu_0 - 1/2 $. Now, consider a non-linear diffusion model $d X_t = \mu_0 X_t^{2 \theta-1}...
- 10.6k
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1
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47
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On the supremum of cadlag function with bounded jumps
For a function $f:[0,\infty)\rightarrow\overline{\mathbb{R}}$ that is a cadlag (right-continuous with left limits) define
$$
T_n = \inf\{t\geq 0\mid \left|f(s)-f(0)\right|\geq n\}.
$$
Suppose that $|\...
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2
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Triangle Inequality for this metric
Let $(X,\varrho)$ be a metric space, and define $d:X\times X\to \mathbb{R}$ by
$$d(x,y)=\inf\lbrace\varrho(x,z)+\varrho(z,y):z\in X\rbrace$$
Show that $d$ is a metric on $X$.
I have all the axioms but ...
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