For questions on infima.

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essential infimum

Let $\Omega$ be a bounded domain in $\mathbb{R}^{N}$, if $u\in L^{\infty}(\Omega)$ and we want define the infimum of $u$ we write $\mathrm{ess}\inf_{x\in\Omega} u(x)$ but if $u\in ...
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Infimum and supremum of a set between 0 and 1

I am a little confused on what the infimum and supremum would be for the set S of all rational number between (0,1) not including 0 and 1. If 0 and 1 were included the answer is quite obvious.
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1answer
39 views

Infimum of absolute values versus absolute value of infimum

Let $A\subseteq\mathbb R$. Is there a nice proof of the inequality $\displaystyle\inf_{a\in A} |a|\le|\inf_{a\in A} a|$? The only proof I know is, though not very difficult, annoying because it ...
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0answers
23 views

Determine the Law of $F^{-1}(U)$

If, $F^{-1}(u)=\inf\{x\in\mathbb R:F(x)>u\}$ and $U$ is uniformly distributed in $[0,1]$, what is the law of $F^{-1}(U)$ ? ($F$ is a distribution function of some random variable $X$) How can ...
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3answers
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If $S=\text{{$-\infty,\ldots,0$}}$, does there exist infimum?

In this question What is the difference between minimum and infimum?, the answer of Thomas contains a line that It is a fact that every set of real numbers has ...
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2answers
49 views

Prove the infimum of monotone decreasing set equals its limit

Suppose that the sequence (sn) is monotone decreasing, in other words s1 ≥ s2 ≥ .... In addition suppose (sn) converges to s ∈ R. With these assumptions, prove that the set E = {s1, s2, ...} has an ...
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1answer
53 views

How to prove that the spectral radius of a linear operator is the infimum over all subordinate norms of the corresponding norm of the operator.

I am trying to understand a proof I have seen of the following theorem: $$\rho(A)=\inf_{\|\cdot\|}\|A\|.$$ I understand that to do this, the idea is to show that 1) $\rho(A)\leq\|A\|$ for any norm, ...
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1answer
78 views

$\exists f:\mathbb{R}\rightarrow \mathbb{R},$ continuous, non-constant, with uncountably many extrema?

I couldnt think of any; by intuition I don't think any can exist, but I can't figure out how to prove it. If it existed then the set of extrema would have to be uncountable but I think this might ...
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3answers
147 views

Prove that infimum (A)=0 and that supremum (A)=1 in the following set

$$A=\{\frac{n}{m}:m,n \in \mathbb{Z}^+, m>n\}$$ Now, I know that, as $n$ approaches $0$ from above and as $m$ approaches infinity, $\frac{n}{m}$ gets arbitrarily close to $0$, but my professor ...
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Determine whether the following subsets of $\mathbb{R}$ are bounded.

$A=\{x+\frac{1}{x}:x \in (0,\infty)\}$ $B=\{x^2+xy^2:-2 \leq x \leq 1, -1 \leq y \leq 1\}$ I understand the what it means for a set to be bounded above and below, but how would I go about proving ...
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292 views

Show that: $inf(A+B)=inf(A)+inf(B)$

Let $A,B$ not empty,bounded subsets of $R$ and $A+B=\{a+b:a \in A,b \in B\}$.Show that: $$\inf(A+B)=\inf(A)+\inf(B)$$ That's what I have tried: -Suppose that $x \in A+B \Rightarrow x=a+b,a \in A,b \in ...
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1answer
69 views

Supremum calculation

Calculate $\sup(\sum_{k=n+1}^{\infty}\frac{|x_{k}|^{2}}{4^{k} })$, where $x=(x_{1},x_{2},....)$ is a member of $l_{2}$ and the supremum is take over all $x$ with $||x||= 1$. My intuition says the ...
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0answers
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an example of supremum using relation other than <. like using divides etc??

i want to know some examples of supremum or infimum on a poset using relation, other than $\le$, for example using divides or mod. basiclly i want to know how to find out the supremum of a poset for ...
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1answer
25 views

Infimum of pre-image of continuous function

Let $f$ be a continuous function on $\mathbb{R}_+$ into $\mathbb{R}$. Then for every $n\in\mathbb{N}$ $$\inf\{t\in\mathbb{R}_+\colon f(t)\in [n,\infty)\}=\inf\{t\in\mathbb{R}_+\colon f(t)\in ...
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1answer
56 views

Inequality with infimum and supremum for $A \subseteq \bigcup_{n=1}^{\infty}A_n$

Is it true that if $$A \subseteq \bigcup_{n=1}^{\infty}A_n$$ then $$ \sup A - \inf A \le \sum_{n=1}^{\infty} \sup A_n - \sum_{n=1}^{\infty} \inf A_n \quad (\star)$$ I suppose that it is true (I need ...
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1answer
100 views

Condition For A Set Having A Smallest Element

I am reading the second part of the prolouge of Spivak's Calculus. In the text, he proves the Well-Ordering Principle. Here is a sentence from the book: Suppose that the set A has no least member. ...
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1answer
154 views

$\inf$ and $\sup$ of a set.

Let $n\geq3$ be an arbitrarily fixed integer. Take all the possible finite sequences $(a_{1},...,a_{n})$ of positive numbers. Find the supremum and the infimum of the set of numbers ...
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1answer
22 views

Reasoning about Schnirelmann Density: Proving that $d(C) \ge d(A) + d(B)$

I am taking this argument from Gelfond & Linnik's Elementary Methods in the Analytic Theory of Numbers. They state if for every $n \ge 1$, there exists $m \in [1,n]$ where $C(n) - C(n-m) \ge ...
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1answer
105 views

Bounded functions and infimum/supremum

Question from my homework: ![question] My proof: Let $ S = \{f(x) | x \in \mathbb{R} \} $, since $f$ is bounded it has an infimum & supremum (is this true? I tried to prove it, but have no ...
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1answer
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Question about Schnirelmann Density and Sumset: if $d(A) \ge \frac{1}{2}$ and $d(B) > 0$, wouldn't $d(A+B)=1$

I've been thinking about the Schnirelmann Density and I think that I may still be confused about SumSet and Density. It seems to me that if $d(A) \ge \frac{1}{2}$ and $d(B) > 0$, then $d(A+{B}) = ...
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170 views

Find the supremum, infimum, maximum and minimum

Find the supremum, infimum, maximum and minimum of this set: $$E = \{\frac{2^p}{5^q}:{p \over q} \in (1,2)\text{ and } q > 0\} $$ My thoughts: there is no supremum because we can choose always ...
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1answer
98 views

Prove that the infimum is not attained for a set $M$

Consider $C([0,1])$ with the $\sup$-norm. Let $$N = \bigg\{ f\in C([0,1]) | \int_0^1 f(x)dx = 0\bigg\}$$ be the closed linear subspace of $C([0,1])$ of functions with zero mean. Let $$X = \{ f\in ...
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1answer
55 views

Additive properties of sequences: trying to understand Schnirelmann density

I have started reading Gelford & Linnik's elementary methods in analytic number theory (1965). They define a sequence $A$ of integers as: $$0, a_1, a_2,a_3,\dots$$ where $$0 < a_1 < a_2 ...
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1answer
91 views

Best way to explain how the Infimum and Supremum of this function are obtained…?

I have the function $\;f(x)=\dfrac{x^{(1/2)}}{2+x}\;$ and I know that $\inf(f)$ does not exist and $\sup(f)=2$ but I don't know how to formally show this rigorously? Anyone got a formal way of showing ...
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0answers
303 views

Find the minimum,maximum, infimum and supremum of sets?

If $X$ is the intersection of all the intervals $(1-\frac{1}{n^2},1+\frac{5}{n^3}]$ for $n=1$ to infinity, what is the minimum, maximum, supremum and infimum of $X$? If $Y$ is the intersection of all ...
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1answer
68 views

Question on the definition of infimum?

Thm: Let $K$ be compact metric space and $f:K\rightarrow \mathbb{R}$ a continuous real-valued function. Then $f$ is bounded on $K$ and attains its infimum. Since $K$ is compact and $f$ continuous ...
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203 views

Let $X$ be a compact metric space. If $f:X\rightarrow \mathbb{R}$ is lower semi-continuous, then $f$ is bounded from below and attains its infimum.

Let $X$ be a compact metric space. If $f:X\rightarrow \mathbb{R}$ is lower semi-continuous, then $f$ is bounded from below and attains its infimum. I want to prove this. This is my proof: Since $X$ ...
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1answer
77 views

Question about intervals and infima/suprema

Let $L(E)$ be the set of lower bounds of a set $E$ and $(S, \le)$ a partially ordered set. For each $s \in S$, let $$ \langle s] := \{x \in S \mid x \le s\} $$ and $$ [s\rangle := \{x \in S \mid ...
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Infimum of a set of a sequence of numbers

Consider the sequence $\{ y_{n} \}$ of real numbers such that $\sup\{\left | y_{n} \right |:n\in \mathbb{N}\}=4$. Find $\inf \left\{\frac{\left | y_{n} \right |}{n}:n\in \mathbb{N} \right\}$ Since ...
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2answers
78 views

I need to find the supremum of the set $S$

Let $S:=\{x\ge 0,\sum_{n=1}^{\infty} x^{\sqrt{n}}<\infty\}$; I need to find the supremum of the set $S$. Could any one tell me where to start?
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2answers
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Troubles calculating a set infimum

I have a set like this one: $$A=\left\{\sqrt{x^2+x}-x, x \in \Re\right\}$$ I am trying to calculate the infimum, which seeing a plot drawn with wolfram has been revealed to be zero (for x=0). ...
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1answer
144 views

Lim inf of sum of functions

When is the lim inf of sum of two real valued functions equal to the sum of their individual lim inf? That is, I am looking for condition on $f$ and $g$ under which $\liminf\limits_{x \rightarrow ...
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How to obtain the infimum of this inequalities?

Let $A$ be the family of functions $f(z)=z+a_2z^2+\cdots$ that are analytic in unit disk $D:\{z:|z|<1\}$ and $S$ is the subfamily of functions that are univalent in $D$. $R(a)$ is the subfamily of ...
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73 views

closest point property of subset of Hilbert space - what are the conditions for existence of inf?

I'm proving the closest point property of a subset of a Hilbert space, ie: $$H$$ is a Hilbert space with a norm generated by the inner product and so on. $$h\in H$$ is a point in H $$M\subset H$$ M ...
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$\inf A = -\sup(-A)$

Let $A$ be a nonempty subset of real numbers which is bounded below. Let $-A$ be the set of of all numbers $-x$, where $x$ is in $A$. Prove that $\inf A = -\sup(-A)$ So far this is what i have ...
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1answer
73 views

Stability under supremum of sets of social choice function with single peaked preferences

Here is a question emerging from reading Moulin, H. (1980). On strategy-proofness and single peakedness. Public Choice, 35(4), 437–455. The setting is as follows: A non-empty finite set of ...
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1answer
118 views

Finding the supremum of the following set [duplicate]

I am stuck on the following problem: Let $P=\{x \in \Bbb R: x\ge 0,\sum_{n=1}^{\infty}x^{\sqrt n}< \infty\}$.Then what is the supremum of $P$? Can someone help me out by providing some ...
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1answer
65 views

Compute infimum

I want to compute the following infimum: $$ \inf\limits_{x_1,\ldots,x_n \geqslant 0} \dfrac{x_1 y_1 + \ldots + x_n y_n}{(a_1 x_1^\alpha + \ldots + a_n x_n^\alpha)^{\frac 1 \alpha}} $$ where $y = ...
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An example of a function which satisfies $\sup_{y}\inf_{x}f(x,y)<\inf_{x}\sup_{y}f(x,y)$

Let $f:X\times Y\to \mathbb{R}$. I could not come up with an example that satisfies $$\sup_{y\in Y}\inf_{x\in X}f(x,y)<\inf_{x\in X}\sup_{y\in Y}f(x,y).$$ Any help would be appreciated. Thanks!
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1answer
104 views

Question about the infimum of $|f(x)-x|$ , where $f(x)=x$, $x$ is a fixed point of the nonlinear equation.

I am trying to check if the following property holds for fixed points: Suppose: $ f(x)= x $ is given, with solution $x = \theta \gt 0 $ I would like to show : $ \forall \epsilon \in (0,1), ...
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1answer
52 views

infimum's basic properties in optimization problem

This problem is in Optimizing over some variables slide of Convex Optimization problem. I have a question about basic assumption in this textbook. $$ \inf_{x,y} f(x,y) = \inf_{x} g(x), where, ...
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96 views

finding infimum of maximum of two functions

Let $r\geq 1$. Let $C>0$ be a constant. For $x\in R, x>0$ Find the following infimum: $$ \inf_{x}\max\left\{\frac{(1+x)^r}{x}; C\frac{1+x}{x}\right\}. $$
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286 views

Show $\sup{A}-\inf{B}=\sup\{a-b:a\in A, b\in B\}$

Let $A, B \subset\mathbb{R}$ be bounded sets. Show $$\sup{A}-\inf{B}=\sup\{a-b:a\in A, b\in B\}$$
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1answer
220 views

Infimum and limit

I was having trouble with the following question. Any help would be highly appreciated. Let $A$ be the set of K-dimensional vectors with non-negative components. Let $B$ be the set of K-dimensional ...
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1answer
731 views

Inequalities related to infimum and supremum

Let $f,g: A \rightarrow \mathbb{R}$ be integrable functions on a closed rectangle $A \subset \mathbb{R}^n$. Let $P$ be a partition of $A$ and $S \in P$ a sub-rectangle. Show that: $m_S(f+g) \geq ...
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1answer
33 views

$f(x) = \inf_{y \in Y} c(x,y) - \inf_{\xi \in X} c(\xi,y) - f(\xi) \Rightarrow f$ is upper semicontinuous

Let $X, Y$ be metric spaces. Given $c: X \times Y \mapsto \mathbb{R}$ continuous, define $$ f(x) = \inf_{y \in Y} \left( c(x,y) - \inf_{\xi \in X} (c(\xi,y) - f(\xi)) \right).$$ Then is $f$ upper ...
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1answer
317 views

Proof that the infimum of a given set $E$ is unique

Prove that a lower bound of a set might not be unique but the infimum of a given set is unique. Attempt: Consider some $E \subset \mathbb{R} $ such that $E \neq \emptyset$. $E$ is bounded below ...
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3answers
485 views

Supremum/Infimum proof

Assume that $\inf(A)>0$ and let $A'=\left\{\frac{1}{x} : x\in A\right\}$. I need to show that $\sup(A') = \dfrac{1}{\inf(A)}$. I think this is quite simple, $\sup(A')$ must be $\dfrac{1}{\inf(A)}$ ...
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1answer
125 views

Infimum of a Hilbert space inner product

This is exercise 5.11 in Brezis's Functional Analysis, Sobolev Spaces, and PDEs. Let $H$ be a Hilbert space, and let $M \subset H$ be a nonzero closed linear subspace. Let $f \in H$, $f \notin ...
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1answer
43 views

Trying to prove $\text{Hdim}(\bigcup X_i)=\sup_i \text{Hdim}(X_i)$

Suppose $X=\bigcup_i X_i$ is a countable union. I'm trying to prove a statement which wikipedia says follows directly from the definition of Hausdorff Dimension: ...