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

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Supremum of all y-coordinates of the Mandelbrot set

Let $M\subset \mathbb R^2$ be the Mandelbrot set. What is $\sup\{ y : (x,y) \in M \}$? Is this known? To be more descriptive: What is the supremum of all y-coordinates of all black points in the ...
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2answers
25 views

On the definition of Scott Continuity

I somewhere encountered the concept of "Scott Continuity" as follows. Let $P,Q$ be partially ordered sets; a function $f:P\to Q$ is Scott continuous if it preserves directed suprema, i.e. for all ...
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53 views

Is my proof that $\inf A=-\sup (-A)$ correct?

From baby Rudin (exercise 5 in chapter 1): Let $A$ be a nonempty set of real numbers which is bounded below. Let $-A$ be the set of all numbers $-x$, where $x\in A$. Prove that $$\inf A=-\sup (-A).$$ ...
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1answer
25 views

How do I show that $\sup_{0\leq x\leq1}|g(x)| \geq \int_0^1|g(x)|dx$

I want to show that $\sup_{0\leq x\leq1}|g(x)| \geq \int_0^1|g(x)|dx$ for real-valued $g$ that is continuous for $0\leq x\leq1$ . Is it enough to say that $|g(x)| \leq \sup_{0\leq x\leq1}|g(x)|$ ...
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1answer
24 views

Supremum of sum of two sequences: $\sup (x_n+y_n) \le \sup x_n + \sup y_n$

Prove that $\sup\{x_n+y_n\}\leq \sup\{x_n\}+\sup\{y_n\}$, if both sups are finite. Furthermore, prove that $\limsup\{x_n+y_n\}\leq \limsup\{x_n\}+\limsup\{y_n\}$ if both limsups are finite.
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Proof sup(c*S)=c*supS [on hold]

I be glad if someone could prove : Assume S, is not empty, and is upper bounded. Assume C>0. $$\sup(c S)=c \sup S$$ Any help will be appreciated.
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2answers
19 views

Find exact upper and lower bounds of set

Heloo, i have a set A={x-x^2>0} x belongs to rationals Need to find exact lower and upper bounds of this set. Tried to draw a graphic but it didnt worked. Any hints or solutions will be ...
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1answer
34 views

what is the difference between bounded and convergent?

I know that bounded means to have an upper or lower bound. Let $E \subset \mathbb{R}$ be nonempty. The set $E$ is said to be bounded above if and only if there is an $M \in \mathbb{R}$ ...
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3answers
61 views

The supremum of rationals that are less than a given number is equal to that number

I have the following theorem to prove. Given a real number $a$, define the set $S$ such that $S = \{x \in \mathbb Q: x < a\}$. Show that $a = \sup S$. My attempt at a proof is as follows ...
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0answers
9 views

What is the process of finding supremum/infimum of a continuous set?

For example given a set, for constant $k\in \mathbb{N}$ $$ \{ (1 + x^2)^{-k} \ | \ x\in [-1,0) \cup (0,1] \} $$ The supremum is $1$ ( the limit as $x$ approaches $0$), but how does one derive this ...
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1answer
27 views

Deduce supremum of Set A

Define $$Set A=\{1-\frac 1n\ | n \in\ N\}$$A. Deduce sup Ab. Use the quanitifier definition of supremum to prove your conjecture in part (a).My attempt at the solution: I believe sup A is 1?
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2answers
60 views

Let $A,B \subset \mathbb{R}$. Show that $\sup(A \cup B) = \max\{\sup A, \sup B\}$

Let $A,B \subset \mathbb{R}$. Show that $\sup(A \cup B) = \max\{\sup A, \sup B\}$ Here's what I did so far: Let $\sup A, \sup B$, and $\sup (A \cup B)$ denote upper bounds of $A,B$, and $A \cup ...
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1answer
86 views

For any $n \in \mathbb{N}$ there exists a minimal element in the set $ \{ x\in \mathbb{N}\mid n< x \}$

Problem from V. Zorich Analysis textbook: For any $n \in \mathbb{N}$ there exists a minimal element in the set $ \{ x\in \mathbb{N}\mid n< x \}$ namely $\min\{x \in \mathbb{N}|n<x\}=n+1$* ...
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1answer
48 views

If $A$ is a subset of $B$, then $\sup A \leq \sup B$

Let $A$ be a subset of $B$, where $A$ is nonempty. How can I show that $\sup A \leq \sup B$? My attempt I said let element $a$ be in $A$, which means $a$ is in $B$ because $A$ is a subset of $B$. I ...
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3answers
50 views

Help with prove of inf.

I need to prove that: $$\inf\{\frac{1}{3}+\frac{3n+1}{6n^2} \Big| n\in\mathbb N\}=\frac{1}{3}$$ I get stuck with my proof, I'll write it down. $$n\geq1$$ $$3n\geq3$$ $$3n+1\geq4$$ ...
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1answer
48 views

Preperation for a test: Show that inf($\frac{1}{n}$)=0 . Please check if what I have is correct

We are given the following definition: If a sequence $(a_n)$ is bounded from below then it has a greatest lower bound for the sequence called a $\textbf{infimum}$. $m=$ infimum of $(a_n)$ if i) ...
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1answer
59 views

Clarification needed: $\inf (A+B) = \inf A + \inf B$

Let $A$ and $B$ be nonempty bounded subsets of $R$ and let $A+B$ be the set of all sums $a+b$ where $a\in A$ and $b\in B$. Prove $\inf(A+B)=\inf(A)+\inf(B)$ My attempt: Since $A$ and $B$ are ...
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31 views

Properties of Sup, Inf on sets.

I understand this proof is a bit long-winded, but I am only concerned with it is correct or not. It seems sound to me. Claim: If $A,B \subset \mathbb{R}$ and are non-empty, $a \leq b, \forall a \in ...
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2answers
68 views

$x+y\sqrt{2}$ infimum ($x,y\in \mathbb{Z}$)

I've looked for help with this question but I have not found anything, I hope this is not a duplicate. Define the set $A=\{\mid x+y\sqrt{2}\mid \ : x,y\in \mathbb{Z}\ \mbox{and} \mid ...
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22 views

Existence of a function satisfying all the given infima

Given a function $f : \mathbb{R} \to \mathbb{R}$, we can compute its infimum over $A$ for all the Borel measurable $A \subset \mathbb{R}$. I am wondering when we can deduce in the other direction, ...
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3answers
174 views

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
47 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
82 views

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
69 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
71 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
80 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
210 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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70 views

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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2answers
429 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
87 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
23 views

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
26 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
60 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
121 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
157 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
24 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
120 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
21 views

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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2answers
208 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
106 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
57 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
97 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
315 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
70 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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1answer
213 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
78 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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0answers
68 views

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
81 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
84 views

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). ...