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

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47 views

An exercise on uncountable subsets of $[0,1]$ [on hold]

I am stuck on how to prove these three questions, or even how to draw the sets so I can see where they overlap. Any help would be appreciated! Let $C\subseteq [0,1]$ be uncountable. Show there ...
0
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1answer
17 views

supremum of a function in a normed space

Let $X$ be a normed space and $A \subset X$. Prove that: $$\sup(f(A))=\sup(f(cl(A))=\sup(f(conv(A)),$$ where $f \in X^{*}$. For the first equality, I thought to prove it by double inequality: Let ...
1
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0answers
24 views

Showing that the inferior integrals of two functions are equal.

Let $g:[a,b]\to\mathbb{R}$ Riemann integrable and put $h(x)=g(x)$ if $x$ is rational and $h(x)=g(x)+1$ if $x$ is irrational. Show that the inferior (Darboux) integral of $h$ is equal to the integral ...
0
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1answer
36 views

Supremum of $\cot(\pi z)$ where $z$ is on circle with radius $n+1/2$

I try to estimate the supremum of $|\cot(\pi z)|$ and where $z=(n+1/2) e^{i t}$, $n\in\mathbb N$ and $t\in[0,2\pi)$. I should be a constant. So far I did by wiriting it in exponential form and ...
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1answer
29 views

Infimum and supremum of two variable function [on hold]

How can I find the infimum and supremum in $\mathbb{R}^{2} $ of this function $$ f(x,y)=(2x^2+y^2-1)(x^2+y^2-1)+1 $$? Thanks EDIT: Forgive me if I did not add my thoughts but I did not know where to ...
4
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1answer
58 views

How to prove that for all $k\in\mathbb N$, $h(kx)=kh(x)$ and $h(x+y)\le h(x)+h(y)$?

Suppose $X$ is a commutative monoid and $f:X\to\mathbb R\cup\{\infty\}$ a function and $$g(x)=\inf\left\{\sum_{i=1}^nf(x_i)~\middle\vert~\sum_{i=1}^nx_i=x,n\in\mathbb N\right\}$$ ...
2
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2answers
23 views

Supremum not in interior of set $A$

If $ A \neq \emptyset $ and is bounded above and $ s = \sup A $ then $ s \in \bar A $ and $ s \notin \operatorname{int}A$ I've got the first part, showing that $s$ is a limit point of $A$. Thus, $ s ...
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3answers
33 views

Why is $\sup_{x∈[0,1]} {|p'(x)|} ≤ A_d\sup_{x∈[0,1]}{|p(x)|}$ for all polynomials $p$ of degree at most $d$?

How can one prove that for any positive integer $d$, there is a constant $A_d < 0$ such that $$ \sup_{x∈[0,1]} {\lvert\, p'(x)\rvert} ≤ A_d\sup_{x∈[0,1]}{\lvert\, p(x)\rvert}, $$ for all ...
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1answer
25 views

Proving infimum and supremum using epsilon definition.

I came to the conclusion that sup$(A)= 3$ and inf$(A)=2$, but I am stuck at part $(a)$ showing it bounded above and below. How do I solve the rest of $(a)$ and $(b)$ using the epsilon definition.
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0answers
40 views

Is this supremum infinite or finite?

Given two numbers $x\in (0,1)$ and $y\in (0,1)$, think of the expression $\min_{n\ge 1} \frac{1-x}{(1-xy)y^{n-1}(1-x^n)}$. Does the supremum of this expression, namely, $\sup_{x,y\in ...
0
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0answers
32 views

Is it ok to use $x$ and $-x$ as counterexample for $L_{f+g}(M)\le L_f(M)+L_g(M)$?

Let $P$ be partition of $[a,b]$. How to give an counterexample to $L_{f+g}(P)\le L_f(P)+L_g(P)$ and $U_{f+g}(P)\ge U_f(P)+U_g(P)$? Use $f(x)=x$ and $g(x)=-x$ and restricting domain to $(0, 1]$. In ...
0
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2answers
39 views

How to prove that if $f$ is integrable, then $\forall \epsilon >0, \ \exists$ partition $M\in [a,b]$ such that $U_f(M) - L_f(M)\lt\epsilon$?

Here is my proof: Since $f$ is integrable, $\overline{I}_a^b(f)$=$\underline{I}_a^b(f)$. However, it is also a fact that $L_f(M) \leqslant \underline{I}_a^b(f) \leqslant \overline{I}_a^b(f) ...
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0answers
18 views

When the supremum of a real sequence is finite and not attained, it coincides with the limsup

I'm having a bit of a problem with an exercise I have to make. In the exercise we are given the sequence $(s_n)$, which is a sequence of reals. Furthermore, we are given that $m=\sup\{s_n|n \geq ...
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0answers
30 views

Why can we calculate the supremum of operator norm over unit circle?

I know that to check whether a linear operator is continuous or not we have to check if the operator norm is bounded. $$T: V\to W$$, $$\vert\vert \ T \vert\vert= \sup_{f \in V}\frac{\vert\vert \ Tf ...
2
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2answers
97 views

Will every subset of $R$ that is not bounded above contains a sequence that diverges?

Question: (a) Prove that every subset of $R$ that is not bounded above contains a sequence that diverges to infinity. (b) Prove that every unbounded subset of $R^d$ contains a sequence ($x_n$) with ...
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1answer
21 views

Supremum Infimum of Norm

Let $A\in\mathbb{R}^{n\times n}$ be an invertible matrix and $\mathbf{x}\in\mathbb{R}^n$. I am trying to prove that ...
2
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1answer
46 views

Spivak's Calculus exercise related to $\sup$ and $\inf$

This is the exercise 4.b of chapter 8 from Spivak: Suppose that $f$ is continuous on $[a,b]$ and that $f(a)<f(b)$. Prove that there are numbers $c$ and $d$ with $a \leq c <d \leq b$ such that ...
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1answer
30 views

Finding a relation between two sets

I am using the textbook Elementary Analysis. I am having trouble solving the following problem: Let $A$ be a set of real numbers and let $B=\{-x \mid x \in A\}$. Find a relation between $\max(A)$ and ...
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1answer
48 views

Does $\sup AB\ge \sup A \sup B$ always?

I know that if $A,B$ are sets of nonnegative numbers, we have that $\sup AB=\sup A \cdot \sup B$, but what happens in the general case, what conditions on $A,B$ do I need for $\sup AB$ to exist? ...
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0answers
30 views

Prove $\inf(-x) = -\sup(x)$

Suppose S is bounded above, Prove $\inf(-x) = -\sup(x)$ I've reckon there are similar proofs here with negative signs in different ways, but I want to get some proof verification as well as some ...
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5answers
72 views

Proving $x<y \implies n^{x}<n^{y}$, for n>1. $x,y \in \mathbb R$

I think I'm supposed to use the lowest upper bound property but I don't even know where to construct a set to start the problem.
0
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1answer
31 views

Proof of $\sup \epsilon x = \epsilon \sup x$

Suppose S is a non empty set of real numbers, and suppose S is bounded above, and that $\epsilon > 0$, Prove $\sup \epsilon x$ = $\epsilon\sup x$ My take so far: $sup S = B$, then $B$ is an ...
0
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1answer
28 views

Supremum and infimum of continuous functions on $[a,b]$

Let $\{ f_\alpha (x) \}_{\alpha \in I }$ a family of continuous functions from $[a,b]$ to $\Bbb{R}$ ($ a<b $). Now we know because $[a,b]$ is compact then every $f_\alpha$ is uniformly continuous ...
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votes
2answers
38 views

Find infimum of this set

I need to find infimum of this set $$ \left\{ \frac a {b+c} + \frac b {a+c} + \frac c {a+b} : a,b,c \in \mathbb R^+\right\}$$ Some hint would be helpful
0
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1answer
48 views

Any example for lower integral $\underline{I}_a^b (s)+\underline{I}_a^b (t)\ne \underline{I}_a^b (s+t) $ for two bounded functions $s$ and $t$?

Any example for $\underline{I}_a^b (s)+\underline{I}_a^b (t)\ne \underline{I}_a^b (s+t) $ for two bounded functions $s$ and $t$? Since it is true that $L_f(M)+L_g(M)\le L_{f+g}(M) $ for partitions M, ...
0
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0answers
30 views

The convergence of series of independent random variables

Let $\{a_n\}$ be a sequence of complex numbers and let $\{A_n\}$ be a non-descreasing sequence of positive numbers, tending to infinity. We assume that $\sup\limits_{n\ge 1} ...
3
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1answer
48 views

Let $\mathbf S = \sin(n)$, where $n \, \epsilon \, \Bbb N$. Prove $ \sup \mathbf S = 1$ and $ \inf \mathbf S = -1$

Let $\mathbf S = \sin(n)$, where $n \, \epsilon \, \Bbb N$. Prove $ \sup \mathbf S = 1$ and $ \inf \mathbf S = -1$. I understand that the $\sin$ function for real entries is normally bounded by $1$ ...
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2answers
29 views

Show that the infimum of the set $\{(3n+4)/n\}$ is $3$, using the Archimedean property

Let E = $\left\{ x|\exists n \in N (x= {\frac{3n+4}{n}})\right\} $. Show that Inf E =3. I need help understanding the last step involing the archimedean property. We have 3 as a lower bound : ...
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2answers
39 views

Inequality for product of integrals [closed]

Consider two real-valued functions $g: \mathbb{R}\rightarrow \mathbb{R}$ and $f:\mathbb{R}\rightarrow \mathbb{R}$. Is this relation true: $$ \int_{\mathbb{R}}g(x)f(x)dx\leq \sup_x ...
0
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0answers
23 views

Notation question VaR and CVaR

Given that I am dealing with a discrete distribution, what exactly does the following notation mean: $\mathbb{E}[X\mathbb{1}_{\{ X \geq x_q\}}]$ where $\mathbb{1}$ is an indicator function. How do I ...
0
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1answer
54 views

Is it true that $L_{f+g}(M)\le L_f(M)+L_g(M)$ if $M$ is a partition on [a,b] and $f, g$ are two bounded functions?

Here I have a bit confusion, if $f$ is positive and $g$ is negative, does that mean the total lower sum is $L_f - |L_g|$ or is it the absolute value? If $f$ and $g$ are both positive, how do you know ...
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1answer
83 views

How to prove that $f$ is integrable if $\forall \epsilon, \ \exists$ partition $M\in [a,b]$ such that $U_f(M) - L_f(M)\lt\epsilon$?

In order to let $f$ be integrable, its lower integral,$\underline{I}_a^b(f)$, has to equal upper integrable, $\overline{I}_a^b(f)$. I suppose that we can claim that because $\forall \epsilon, U_f(M) ...
0
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1answer
36 views

How to prove that a real number is the lower integral of the function based on the following assumption?

Let $M$ be a real number, If $1.\ \forall\epsilon\gt0,$ there exists a partition $P$ of $[a,b]$ such that $M-\epsilon \lt L_f(M)\le M.$ $\ \ \ 2.\ L_f(M)\le M$ for every partition P of $[a,b] $ . ...
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6answers
52 views

Supremum proof simple

I got stuck on this problem and can't figure it out, I hope somebody can help me, I also wrote my attempt. Thanks in advance!! Question: Let $(a_n)$ be a convergent sequence in $\mathbb{R}$. $a_n ...
0
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0answers
34 views

Does the following relation hold?

Suppose $f$ and $g$ are Schwartz functions. Does the following result hold? $\sup_x|f(x)g(x)|=\sup_x|f(x)|\sup_x|g(x)|$ In general, for what class of functions this is valid ? Or is it that this ...
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2answers
48 views

Example closed sets for which $\inf(A+B) > \inf A+\inf B$

A known inequality states: $$\inf(A+B) ≥ \inf A+\inf B$$ Now what are example (closed) sets of the "$>$" case?
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0answers
24 views

$supA ≤ infB$ $\implies$ $\neg$ of $supA=infB$ is $supA > infB$?

I'm a bit confused by the reasoning in this paper: http://www.feferraz.net/files_/lista/mat206-l1e14.pdf For two non-empty real subsets s.t. $a≤b$ $\forall a\in A, b\in B$, it first proves $supA ≤ ...
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2answers
22 views

Supremum proof real analysis

hope somebody can help me. I have a feeling it is a simple question, but I just cannot figure it out. Suppose A and B are two non-empty bounded subsets of $\mathbb{R}$ such that $\sup A =\sup ...
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3answers
61 views

one real analysis problem to prove the existence of a sequence and limit of a sequence

Question: Suppose that S is a nonempty subset of R that is bounded above and put s = sup S.Show that there is a sequence (xn) such that xn ∈ S for all n and lim as n apporaches infinity xn = s. Hi, ...
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2answers
24 views

Supremum of non-empty subset of $\mathbb{R}$

Let A be a non-empty subset of $\mathbb{R}$ that is bounded above and put $s=\sup A$ Show that if $s\notin A$ the the set $A\cap (s-ε,s)$ is infinite for any $ε>0$ This has to be solved using ...
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1answer
29 views

How does the superimum of lower sums of partitions work?

The definition of integral defines lower integral of $f$ on $[a,b]$, $\underline{I}_a^b (f)$, as the supremum of the lower sums of all partitions. But isn't the lower value of $f(x)$ infimum? how ...
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0answers
15 views

Exchanging supremum and expectation

Consider a real-valued random variable $X_i$ defined on the probability space $(\Omega, \mathcal{F}, P)$. Let $m(X_i;\theta)$ be a random function which depends on the parameter $\theta \in \Theta ...
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1answer
122 views

Example of a chain without a supremum in a non Archimedean ordered field

I give here the example of a non-Archimedean ordered field. I know that the field is not order complete. What is a simple example in that field of a chain without a supremum?
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1answer
31 views

Supremum of set in more than 2 dimensions: i.e. supremum of set in $\mathbb{R}^n$

Let us consider $\mathbb{R}^n, n\geq 2$. For simplicity, let's say $n=2$ (but I'm wondering about when $n>2$ as well). lets consider the set $$ Z=\{ (x,y)\mid x+y\leq 2\} $$ Does this set have a ...
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1answer
70 views

Suprema and infima in real numbers

I am having troubles when finding suprema and infima of sets. Could the supremum exist when the set is bounded from above? For example, $A=\{x\in \Bbb R \mid x^2<5\}$, the supremum is $\sqrt5$, ...
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1answer
42 views

Uniform Integrability and relation to $L^p$ for $p>1$

Let $X_n$ be a martingale. Then we know that for $p> 1$ the conditions $\sup_n E[|X_n|^p] < \infty$ and $E[\sup_n |X_n|^p] < \infty$ are equivalent. For $p=1$ this does not hold, because ...
0
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0answers
22 views

Does argsup function satisfy a property of the supremum

Let $X$ be a compact set of $d\times d$ matrices, and let $f\in C(\overline{\Omega})$, and $u\in C^2(\overline{\Omega})$. Define $A(x)=\operatorname{argsup}_{W\in ...
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0answers
25 views

Brownian motion and sup of a Brownian motion

I am stuck with the following problem: let $B_t$ be a standard Brownian motion and let $S_{t}:=\sup_{0 \leq s \leq t} B_s$. Prove that for every $\lambda \geq 0$ and $\mu \leq \lambda$, ...
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2answers
31 views

Show that $\underset{x\in X}{\sup}\left(\underset{y\in Y}{\sup}f(x,y)\right) = \underset{(x,y)\in X\times Y}{\sup} f(x,y)$

I have the following problem I don't know how to start. Prove that: $$\underset{x\in X}{\sup}\left(\underset{y\in Y}{\sup}f(x,y)\right) = \underset{(x,y)\in X\times Y}{\sup} f(x,y),$$ ...
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1answer
37 views

How to prove the following properties of infimum and supremum involving the union and intersection of the sets $A_k$

I am reading a book on probability theory and I have troubles understanding why the following holds $$ \sup_{k \ge n} A_k = \bigcup_{k\ge n} A_k $$ $$ \inf_{k \ge n} A_k = \bigcap_{k\ge n} A_k $$ ...