For questions on suprema and infima. Use together with a subject area tag, such as (real-analysis) or (order-theory).

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27 views

On the integration of a Lebague measurable function [on hold]

Sincerely need help on this question, anyone who has any ideas please don't hesitate to tell me :)
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1answer
16 views

Partial ordered sets and supremum.

Let $A,B$ subsets of a partial ordered set $(X,\succeq)$, such that $A\subseteq B$. Suppose that $sup(B)$ exists, Do $sup(A)$ exists? I could prove that all upper bound of $B$ is an upper bound of ...
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0answers
29 views

Infimum of $f(x)=x$ for $x\geq 1$ and $x\leq 0$

I have a query and maybe some of you might help me. I have in my notes that the infimum of $f(x)=x$ is $-\infty$. And that the infimum of $f(x)=x$ $x\geq 1$ and $x\leq 0$ is $+\infty$. I do not ...
0
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1answer
27 views

Query about non-singular transformation of vectors

Suppose we are given a probability function, P (x^T (Y-z)≥0) , where ‘x’ is a vector, ‘Y’ is a random variable and ‘z’ is a known value. Now, suppose, we make a non-singular transformation w=Ax, ...
1
vote
1answer
48 views

If an element $a \in A$ is an upper bound for $A$, then $a = \sup A$.

Prove that if an element $a \in A$ is an upper bound for $A$, then $a = \sup A$. To prove this, do I need two cases and follow the definitions of the infimum and supremum?
1
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1answer
31 views

Infimum of $f(x): (0,\infty) \to \mathbb R$, $f(x) = \ln(e^x-1)+\frac2x-x$

Calculate infimum of $f(x): (0,\infty) \to \mathbb R$, $f(x) = \ln(e^x-1)+\frac2x-x$ I calculate derivative $$f'(x)= (\ln(e^x-1)+\frac2x-x)' = \frac1{e^x-1}e^x-\frac2{x^2}-1 = ...
0
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1answer
15 views

Supremum Proof Questin

$A_1$, $A_2$, $A_3$.... are a collection of nonempty sets, each bounded above. I'm asked to find a formula for sup($A_1$$\cup$$A_2$) and then to extend this to $\bigcup^{n}_{k=1}$$A_k$. For the ...
1
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1answer
35 views

Every closed subset of $\mathbb R^n$ has a point that minimizes the distance to a given point $p\in\mathbb R^n$

Let $p\in\mathbb R^n$ and $\|\cdot \|$ the Euclidian norm. Show that if $K\subset \mathbb R^n$ is a close set, then $$\exists a\in K: \forall x\in K, \|a-p\|\leq \|x-p\|.$$ Since $\|x-p\|\geq 0$, ...
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2answers
36 views

Prove the following properties of sequence

Define $$L = \limsup_{k \rightarrow \infty}a_k =\inf_j(\sup_{k\geq j}a_k).$$ Prove that if $(a_k)$ and $(b_k)$ are sequence of real numbers then $$\limsup(a_k + b_k) \leq \limsup a_k + \limsup b_k.$$ ...
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1answer
20 views

Wheeden-Zygmund exercise

Define $\limsup_{k \rightarrow \infty}a_k$ and $\liminf_{k \rightarrow \infty}a_k$ by $$\limsup_{k \rightarrow \infty}a_k = \lim_{j\rightarrow \infty}b_j = \inf_{j}\{\sup_{k\geq j}a_k \} $$ ...
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2answers
42 views

Prove that if $\lim_{n\to \infty} a_n=l$ then $\lim \sup a_n=\lim \inf a_n=l$

Let $(a_n)_{n\in\mathbb{N}}\subset\mathbb{R}$, a bounded sequence. For each $n\in\mathbb{N}$, we have $A_n=\{a_k:k\ge n\}$. Let $\lambda_n=\sup A_n$ and $\beta_n=\inf A_n$.So we have $(\lambda_n)$ and ...
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4answers
34 views

Prove that the least intercept made on the tangents to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ by the axes is $a+b$.

Prove that the least intercept made on the tangents to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ by the axes is $a+b$.Also find the point of contact of the corresponding tangent. I tried.Let ...
0
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1answer
33 views

Show that for each $a>0$ the function $e^{-ax}x^{a^2}$ has a maximum value, say $F(a)$, and that $F(x)$ has a minimum value $e^{-e/2}$

Show that for each $a>0$ the function $e^{-ax}x^{a^2}$ has a maximum value say $F(a)$,and that $F(x)$ has a minimum value $e^{-e/2}$. I differentiated the function $f(x)=e^{-ax}x^{a^2}$ to get ...
0
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1answer
32 views

What is the radius of smallest circular disk large enough to cover every acute isosceles triangle of a given perimeter $L$?

What is the radius of smallest circular disk large enough to cover every acute isosceles triangle of a given perimeter $L$? Let $a,a,b$ are the sides of the isosceles triangle whose perimeter is ...
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1answer
38 views

Find the cosine of the angle at the vertex of an isosceles triangle having the greatest area

Find the cosine of the angle at the vertex of an isosceles triangle having the greatest area for the given constant length $l$ of the median drawn to its lateral side. I tried to solve this ...
0
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1answer
20 views

Prove that the greatest area which the $\Delta APY$ can have is $3\sqrt3\frac{a^2}{8}$sq units

From a fixed point $A$ on the circumference of a circle of radius $a$,let the perpendicular $AY$ fall on the tangent at a point $P$ on the circle,prove that the greatest area which the $\Delta APY$ ...
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0answers
27 views

Find the maximum volume of the cylinder.

A cylinder is obtained by revolving a rectangle about the $x-$axis,the base of the rectangle lying on the $x-$axis and the entire rectangle lying in the region between the curve $y=\frac{x}{x^2+1}$ ...
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1answer
22 views

Supremum Proof Question

Let a$<$b be real numbers and consider the set T=$\mathbb{Q}$$\cap$[a,b]. Show supT=b. I can show that b$\geq$x for all x$\in$T and thus an upper bound, but am not sure how to go about showing it ...
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0answers
17 views

The existence of such function satisfies the infinitum condition

Let $t\in [0,+\infty)$ be a real number, I am looking for two functions satisfies: $f$: $[0,1]\to[0,1]$ such that $f(0)=0$, $f(1)=1$, and $f(t)>0$ if $t>0$. $f$ is lower semicontinuous ...
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2answers
31 views

$\left(a-\frac{1}{r^2}\right)\left(b-\frac{1}{r^2}\right)=h^2$

If $ax^2+2h xy+by^2=1$,prove that the maximum and minimum values of $x^2+y^2$ are given by the values of $r^2$ satisfying the relation $\left(a-\frac{1}{r^2}\right)\left(b-\frac{1}{r^2}\right)=h^2$ ...
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2answers
43 views

Let $(s_n)$ be a convergent sequence of real numbers such that $s_n \neq 0$ for all $n \in \mathbb{N}$ and $\lim_{n \to \infty}s_n=s\neq 0$.

Prove that $\sup \{\frac{1}{|s_n|} : n \in \mathbb{N}\}>0$ Any help on getting this proof started would be appreciated. I know it must be related to proving that $\inf \{|s_n|:n \in ...
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1answer
54 views

The least value of the function $f(x)=|x-a|+|x-b|+|x-c|+|x-d|$ [duplicate]

If $a<b<c<d$ and $x\in\mathbb R$ then what is the least value of the function $$f(x)=|x-a|+|x-b|+|x-c|+|x-d|\ ?$$ $f(x)= \begin{cases} a-x+b-x+c-x+d-x & x\leq a \\ ...
2
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0answers
53 views

Proof of unboundedness of a set

Prove that $\{2^n : n \in \mathbb{N} \}$ is not bounded. Proof: By Induction; $2^n \geq n,\forall n \geq 1$. By Archimedean property $\mathbb{N}$ is not bounded above. Assume that $\{2^n : n \in ...
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1answer
19 views

Maximum occurring at all points in a set

Is there any term for "sets where maximum of a mathematical expression in attained"? I just want to know if the set has any specific name. The set is infinite (do not consider discrete points). The ...
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2answers
45 views

Supremum and infimum of $\left\{(n^2+2n+1)^{\frac{1}{n^2}} \mid n \in\mathbb N \right\}$

Task is to find infimum and supremum of $\left\{(n^2+2n+1)^{\frac{1}{n^2}} \mid n \in\mathbb N \right\}.$ I start from calculating derivative of $ f:\mathbb{R} \rightarrow \mathbb{R}$ where $ ...
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1answer
54 views

find the supremum and infimum of $E =\{x \in \mathbb{R} : -1/n \le x \le 1-1/n\}$

![I am not sure how to start and what it the answer of this question please justify this answer of this question.]1 I am not sure how to start and what it the answer of this question please justify ...
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0answers
29 views

Prove that there is an integer $k$ such that $x \lt k \lt y$.

Suppose that $y-x \gt 1$. Prove that there is an integer $k$ such that $x \lt k \lt y$. I found this prove: "Let $l$ be the largest integer such that $l\le x$. Since l is the smallest integer ...
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2answers
44 views

Show that exists maximum of $A_x=\{m\in \mathbb{Z}: m\le x\} $

For every $x \in\mathbb{R}$, we define $A_x=\{m\in \mathbb{Z}: m\le x\} $. What i have done thus far is show that $A_x\neq\emptyset$, because if $x\in\mathbb{Z}, A_x=\{x\}$ and as ...
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0answers
16 views

Can you always find the least (or greatest) element of a half-closed set?

Was reading this proof about the existence of an irrational number between two rationals. http://www.math.usu.edu/~rheal/math4200/class_material/irrationals%20between.pdf What caught my attention is ...
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1answer
41 views

Find min,max, inf, sup (if they exist) of the sequence $x_n=(-1)^n \cdot \frac{\sqrt{n}}{n+1}+\sin \frac{n \pi}{2}$.

We've been given the sequence $x_n=(-1)^n \cdot \frac{\sqrt{n}}{n+1}+\sin \frac{n \pi}{2}$. I have to find the min, max, inf and sup (if they exist), and also find the points of accumulation. I ...
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1answer
22 views

$\limsup_n a_n + \limsup_n b_n \le \limsup_n c_n + \limsup_n d_n$ if $a_n+b_n=c_n+d_n$ and $a_n$ maximum

I already posed some questions on inqualities between superior limits of real sequences, so here there is another one: Let $(a_n)_{n\ge 1}, (b_n)_{n\ge 1}, (c_n)_{n\ge 1}$, and $(d_n)_{n\ge 1}$ be ...
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0answers
30 views

Implicit Fact about Bounded Monotone Sequence converges

If you have a sequence $ a_n $ which is bounded and monontonic increasing the theorem tells us it converges to a limit $ L $ But having looked at the proof is it implicit that this limit $ L= Sup ...
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2answers
17 views

minimum or infimum for countable sets?

Suppose we have a countable set $X$, say $X=\mathbb{N}$, and let $Q \colon X \rightarrow \{0,1\}$ be a function. Is $$\min \{x \in X \colon Q(x)=1\}$$ the same as $$ \inf \{x \in X \colon ...
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0answers
18 views

Conditions for the existence of moments of the supremum of a random variable

let $X_1, X_2,\dots X_n$ denote a sequence of $n$ iid random variables with the first $k$ moments of $F_X$ exist. Under what conditions (if at all) do the first $k$ moments of the random variable ...
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2answers
37 views

Find supremum and infimumm of a set with two variables

$$A= \left\{\frac{m}{n}+\frac{4n}{m}:m,n\in\mathbb{N}\right\}$$ Since $m,n\in \mathbb{N}$, infimum is zero because $m,n$ both are increasing to infinity. Then the supremum is $5$ when $m,n$ are ...
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0answers
18 views

$\liminf_n \min(a_n,b_n)=\min( \liminf_n a_n, \liminf_n b_n)$

Do you have a reference for the following intuitive result? Let $(a_n)_{n\ge 1}$ and $(b_n)_{n\ge 1}$ be two sequence of reals. Then $$\liminf_n \min(a_n,b_n)=\min( \liminf_n a_n, \liminf_n b_n).$$
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0answers
36 views

$\liminf_n a_n = \inf_n a_n$ if $a_n \ge a_m$ when $n\mid m$

I would like to ask a reference for the following very easy result: can someone help? Let $(a_n)_{n\ge 1}$ be a sequence of positive reals such that $a_m \le a_n$ whenever $n$ divides $m$. Then $$ ...
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1answer
24 views

Why is the lagrange dual function concave?

In a book I'm reading it says I'm struggling to understand the last sentence. Why can one conclude concavity from having a pointwise infimum of a family of affine functions?
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3answers
77 views

If $A \subset B$ m then $\inf B \geq \inf A $?

Suppose $A,B$ are nonempty sets such that $A \subseteq B$. I want to show that $\inf B \geq \inf A $. Suppose $x \in A$ arbitrary. We know $x \geq \inf A $. As $x \in B $, We know out previous ...
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2answers
25 views

What is the difference between maximal element and least upper bound?

Maximal element is given as: Let $(P,\leq)$ be a partially ordered set and $S\subset P$. Then $m\in S$ is a maximal element of $S$ if for all $s\in S$, $m \leq s$ implies $m = s$. Least upper ...
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2answers
65 views

Meaning of symbols like $\inf\limits_{\epsilon>0}$

I am very confused at the precise definition of the following symbols. A reference or explanation would be great. $$\Large\inf\limits_{\epsilon>0}\qquad \sup\limits_{\epsilon>0}$$
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1answer
46 views

$\sup(\sup) \leq 2 \inf(\sup)$

Consider a function $ F : [a; b] \times [a; b] \longrightarrow [0, \infty)$ such that $F(x; y) = F(y; x)$ and $F(x; y) \leq F(x; z) + F(z; y)$ for all $x, y, z \in [a; b].$ Prove that $$ \sup_{x \in ...
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1answer
34 views

A property of infimum??

Let $X$ be some space (eg. vector space or Banach space). When is it true that: for any $\epsilon >0$ small, there exists an $f \in X$ such that $$(1+\epsilon) \inf_{g \in X} I(g) \geq I(f)?$$ ...
2
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1answer
25 views

How does this follow from the theorem?[normed linear space]

I have this theorem: Let X and Y be normed linear spaces and let $T:X\rightarrow Y$ be a linear transformation. The following are equivalent: a. T is uniformly continuous. b. T is ...
3
votes
1answer
46 views

Prove that a sequence sup $a_n\ne1$

I have to prove the following: Let $(a_n)$ be a sequence such that $(a_n)<1$. Prove or disprove: If $(\frac{1} {1-a_n} )$ is bounded from above, then $\sup(a_n) \ne 1$. I was thinking ...
2
votes
2answers
60 views

If $\sup \{a_n\mid n\in \mathbb{N}\}=1$ then $\frac{1}{1-a_n}\to\infty$

Suppose $(a_n)$ is a sequence such that $a_n<1$ for all $n$ and $s:=\sup \{a_n\mid n\in \mathbb{N}\}=1$. I want to prove that $\frac{1}{1-a_n}\to\infty$. My initial approach: let $M>0$ and ...
0
votes
1answer
26 views

The supremum of the function $f(x)=\frac{1-\cos Nx}{1-\cos x}$

I have the following function: $f(x)=\dfrac{1-\cos Nx}{1-\cos x}$ Where N is integer. I know the function has Sup when x goes to $2n\pi$ $n\in\mathbb{N}$. But is it possible to show this? Thank ...
0
votes
1answer
20 views

Supremum over separable Banach space of a measurable function is measurable

Let $X$ be a separable Banach space. Suppose that $f:[0,T] \times X \to \mathbb{R}$ is such that $t \mapsto f(t,x)$ is measurable. Is the function $$t \mapsto \sup_{x \in X}f(t,x)$$ also measurable? ...
0
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2answers
62 views

Show that $\sup (A\cdot B)=\max\{\sup A\cdot\sup B, \sup A\cdot\inf B,\inf A\cdot\sup B,\inf A\cdot\inf B\}$

Given nonempty subsets $A$ and $B$ of positive real numbers, define $$A\cdot B=\{z=x\cdot y:x\in A,\,y\in B \}$$ show that if $A$ and $B$ are bounded sets of real numbers, then $$\sup(A\cdot ...
3
votes
1answer
74 views

Boundedness of the function

Let $x\in(0,1)$ and $S_{n-1}=\sum\limits_{k=0}^{n-1}x^k$. Then define $f$ as the following :$$f(x)=\sum_{n=1}^{\infty}\left|\frac{nx^n}{S_{n-1}}-1\right|$$ I need to show that ...