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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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 ...
Yalini Kumaran's user avatar
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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 { ...
lorilori's user avatar
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2 answers
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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 ...
Shishir Oneal's user avatar
1 vote
1 answer
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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 ...
Donglai's user avatar
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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 $v_0 \perp V$...
Anacardium's user avatar
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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 ...
LideR_M's user avatar
0 votes
1 answer
48 views

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+\...
qerty149's user avatar
4 votes
2 answers
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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 ...
random's user avatar
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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 ...
Ryan Hu's user avatar
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2 answers
50 views

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 ...
qerty149's user avatar
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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 ...
Don P.'s user avatar
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1 answer
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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 ...
Lumos's user avatar
  • 101
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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 $...
Skywear's user avatar
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2 votes
1 answer
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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 $\sup$ ...
Luca T. Castrillón's user avatar
3 votes
1 answer
50 views

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 $\{a^...
upanddownintegrate's user avatar
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2 answers
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$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\...
some_math_guy's user avatar
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1 answer
19 views

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, ...
ADotByMyName.'s user avatar
0 votes
1 answer
28 views

(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 ...
some_math_guy's user avatar
2 votes
2 answers
38 views

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 ...
Lumos's user avatar
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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 ...
noballpointpen's user avatar
1 vote
1 answer
103 views

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 $A$. ...
qerty149's user avatar
1 vote
1 answer
41 views

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}...
Marm's user avatar
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0 votes
1 answer
48 views

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 $B=\{b\in \mathbb{R} : b\space is\space an\space upper\space bound\space for\space A\}$ ...
Lumos's user avatar
  • 101
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0 answers
20 views

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}\...
Ratanjit 's user avatar
0 votes
1 answer
69 views

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)...
S.S's user avatar
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0 votes
1 answer
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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 ...
some_math_guy's user avatar
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0 answers
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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{...
Druizr's user avatar
  • 111
2 votes
1 answer
100 views

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 ...
Remu's user avatar
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1 vote
0 answers
62 views

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 (...
Jord van Eldik's user avatar
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25 views

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 ...
upanddownintegrate's user avatar
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67 views

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 $...
FabrizzioMuzz's user avatar
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0 answers
36 views

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 ...
pmoi's user avatar
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0 answers
19 views

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 $\...
infinit1111's user avatar
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1 answer
49 views

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 ...
Seth's user avatar
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1 vote
0 answers
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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, ...
aya_haitham's user avatar
3 votes
1 answer
32 views

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 $\...
daㅤ's user avatar
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0 votes
1 answer
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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 ...
Wallace Wyatt's user avatar
3 votes
1 answer
46 views

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 ...
cento18's user avatar
  • 341
0 votes
1 answer
57 views

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 ...
Halk's user avatar
  • 103
0 votes
1 answer
47 views

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 ...
THIRUMAL 5688's user avatar
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0 answers
21 views

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 $$ \...
Mohan's user avatar
  • 1
6 votes
1 answer
86 views

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 ...
Peter Sampodiras's user avatar
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0 answers
48 views

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 ...
Leudofikus De Ferento's user avatar
1 vote
1 answer
28 views

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 ...
noobatmath's user avatar
0 votes
1 answer
45 views

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} \...
mathrunner's user avatar
1 vote
0 answers
58 views

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 ...
langmai's user avatar
  • 11
9 votes
1 answer
286 views

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}}, ...
psl2Z's user avatar
  • 1,128
1 vote
0 answers
26 views

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}...
Przemo's user avatar
  • 10.6k
0 votes
1 answer
47 views

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 $|\...
AlmostSureUser's user avatar
1 vote
2 answers
47 views

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 ...
Brandon Myers's user avatar

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