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

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2
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1answer
18 views

Upper bound for the infimum of $(K-x)^2 + (T/x)^2$

I have a function $$f(x)=(K-x)^2 + \left( \frac{T}{x} \right)^2$$ where $K$ and $T$ are positive constants and $x>0$. The function $f$ (hopefully) has an infimum, in terms of $K$ and $T$. I ...
1
vote
2answers
28 views

Norm is sup of inner products (proof).

Let $V$ be a vector space with an inner product $\langle.,. \rangle$ and associated norm $|| . ||$ Then: Could I have a proof of this fact?
0
votes
2answers
27 views

How to use the definition of supremum to prove that $\sup{S} = 2$?

Let $S$ $=$ {$2-\frac{3}{n+1} | n\in \mathbb{N}$}. Use the definition of supremum to prove that $\sup{S} = 2$. Here's what I have so far. We want to show that $2$ is an upper bound, and it is also ...
0
votes
1answer
23 views

Inequality with supremum of divergence of vector fields

I would like to write this inequality: $$\sup\left\{ \int_{A \cup (\mathbb{R}^d \setminus F)} \operatorname{div} T \space \rm d m^d : T \in C_0^1(\mathbb{R}^d,\mathbb{R}^d), |T(x)| \le 1 \right\} \le ...
0
votes
1answer
43 views

a question related to supremum and infimum

$T_n^*:=sup\{t|\sum ψ(x_i;t)\gt 0\}$ $T_n^{**}:=inf\{t|\sum ψ(x_i;t)\lt 0\}$ As it's seen in the above figure, $-\infty \lt T_n^{*} \le T_n^{**} \lt +\infty$ Then, how to write the two followings ...
0
votes
3answers
46 views

Supremum of a sequence of functions

I have a problem understanding the function $\sup f_n(x)$ on a domain of convergence. Given a sequence of functions $f_n(x)$ I guess that $\sup f_n(x)$ is the function such that for each x we find the ...
0
votes
1answer
33 views

Pointwise convergence of $c$-concave envelopes as the sequence of cost functions converges pointwise

On Remark 1.12, page 33 of Villani's Topics in optimal transport we find the following problem. It should be easy but it's turning me mad. Let $c:\mathbb{R}^n\times\mathbb{R}^n\rightarrow ...
3
votes
1answer
94 views

Greatest Lower Bound of the Set of Upper Bounds for a Function

I'm currently in the process of reading a paper, and am trying to work through some of the details on my own. In order to ask my question more effectively, I'm going to begin with a little background: ...
0
votes
1answer
29 views

Conditions for equality in the case of calculation of the supremum of $|f|^2$

It is obvious that $$\sup_{[0,T]}{\left(|f|^2\right)}\leq (\sup_{[0,T]}|f|)^2.$$ My questions is when does the equality hold? Progress: If $|f(x)|$ is not the zero function on $[0,T]$, the equality ...
1
vote
2answers
20 views

Image of a convergent sequence in an increasing function

Suppose I have a function $f:[a,b) \to \mathbb{R}$ that is increasing on its domain and a sequence $a_n \subset [a,b)$ such that $a_n \to b$ and $f(a_n)\to l\in \mathbb{R}$. How would I go about ...
0
votes
1answer
18 views

Prove: Monotonic And Bounded Sequence- Converges

Let $a_n$ be a monotonic and bounded sequence, WLOG let assume it is monotonic increasing. $a_n$ is bounded therefore there is a Supremum, $Sup(a_n)=a$, therefore $a_n<a+\epsilon$. On the other ...
0
votes
0answers
23 views

Proof concerning nested intervals property

I have read and believe to understand the Nested Intervals property theorem but am not sure how to approach the following proof. (The theorem it refers to is just the Nested intervals theorem). ...
0
votes
1answer
86 views

The importance of being real

Let $\Sigma$ be a collection of holomorphic, one-to-one function from some simply connected region $\Omega$, which map $\Omega$ into the open unit disc $U$. Fix $z_0 \in \Omega$ and put $$\eta = ...
1
vote
1answer
29 views

If the set $B=\{f(x) : x\in A\}$ has supremum and $C=\{k+f(x): x\in A\}$, then what is $\sup C$?

If the set $B=\{f(x) : x\in A\}$ has supremum and $C=\{k+f(x): x\in A\}$, then what is $\sup C$? Since $C$ is not an empty set and $f(x)\le \sup(B) ⇒ k+f(x)\le k+\sup(B)$. So $C$ is bounded ...
1
vote
2answers
27 views

Monotonous everywhere function

$f: \mathbb R \to \mathbb R,\forall x \in \mathbb R $ $\exists \delta \gt 0 : f$ is non-decreasing on $(x-\delta,x+\delta)$(I call that statement A). I need to prove that $f$ is non-decreasing on ...
3
votes
2answers
51 views

Sup and inf of $n \sin(1/n)$

If $n$ is a natural number then, what is the supremum and infimum of $n\sin(1/n)$? is the question I want to solve. I drew $sin(x)/x$ graph and I think that the supremum is $1$ and infimum is ...
2
votes
2answers
57 views

Calculate $\sup\limits_{x\in(0,+\infty)}\frac{x^2 e^{-n/x}}{n^2+x^2}$

Calculate $$\sup_{x\in(0,+\infty)}\frac{x^2 e^{-n/x}}{n^2+x^2}$$ The derivative is $$\frac{n e^{-n/x} (n+x)^2}{(n^2+x^2)^2}\geq 0$$ then $$\sup_{x\in(0,+\infty)}\frac{x^2 ...
0
votes
2answers
38 views

Limit of monotonic function

I have to prove that if $(x_1 \gt x_2) \Rightarrow (f(x_1) \ge f(x_2))$, then $$\forall a \in \mathbb R \exists L \in \mathbb R \lim_{x \to a^+}f(x) = L$$ I have a feeling that L = $inf_{x \in (a,a+ ...
1
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0answers
30 views

Let A= [0,1] - {1/n │n ∈ N}. Find sup(A), inf(A), min(A), max(A).

My idea of this question is to claim sup(A) and inf(A) exists (and equals a value) and prove by contradiction that min(A),max(A) exists afterwards (and equals sup(A),inf(A)). The issue that I have is ...
1
vote
0answers
23 views

Supremum/Infimum and min/max

(Problem written verbatim) Let A = [0,1] - {$\frac1n$|n $\in$ $\mathbb{N}$}. Find supremum, infimum, maximum, and minimum of A. Justify your answers. I'm not really sure where to start. I tried ...
2
votes
2answers
24 views

Supremums and subsets

Let $A \subseteq B$, where $A$ and $B$ are non-empty sets and $B$ is bounded above. Show that $\sup(A)$ exists and $\sup(A)\le \sup(B)$ (Problem given as written) For clarity, $\sup(x)$ is the ...
0
votes
0answers
20 views

Find sum of supremum and infimum of $\tan(x)^{\sin 2x}$

Given a function $f(x) = \tan(x)^{\sin 2x}$ I have to show that it has infimum and supremum, this was easy. I already proved that $f$ takes the value equal to infimum (respectively supremum) for ...
1
vote
2answers
43 views

What exactly is the meaning of the following $\inf\{ s_n : n > N\}$ and $\sup\{ s_n : n > N\}$

What exactly is the meaning of the following $$u_N = \inf\{ s_n : n > N\} \ \ \text{ and} \ \ v_N = \sup\{ s_n : n > N\}$$ This might seem a stupid question, but I am not understanding the ...
-1
votes
1answer
47 views

Existence of a sequence that converges to infimum of a function

X is a compact subset of $\Re^{n}$. A upper and lower bounded function f is defined that f:X $\rightarrow$ $\Re$. Does a sequence of {$x_n$} always exist so that f($x_n$)$\rightarrow$ $\inf$[f(x)]. ...
0
votes
1answer
87 views

Equality of sets supremum and infimum

Suppose we have a right-continuous function $f$ defined on $[0,\infty)$. I now would like to prove if $\lambda >0$ and $t\geq 0$ we have that \begin{equation} \left\{ \inf \{ s\geq 0 : f(s) \geq ...
2
votes
2answers
60 views

${\rm sup}\ A\cap B = {\rm min}\ \{ {\rm sup} (A), {\rm sup}(B) \} $

Let $A,B\subseteq \mathbb{R}$ be a non-empty intervals and bounded from above. If $A\cap B\neq \emptyset $ prove that it is bounded from above and that $Sup(A\cap B)=min\{sup(A),Sup(B)\}$ ...
0
votes
1answer
56 views

If $A$ has no max and $B$ is finite, then $\sup(A)=\sup(A\setminus B)$

Let $A\subset \mathbb{R}$ be non-empty and bounded from above, and assume it does not have a maximum. Let $B$ be a finite set of real numbers. Prove: $\sup(A)=\sup(A\setminus B)$ ...
1
vote
1answer
23 views

Prove the least upper bound of a set is in the closure of that set

If $S \subset \mathbb{R}$ and $x$ is the least upper bound of $S$, prove $x$ is in the closure of $S$. I think this means I either have to show that $x \in S$ or $x$ is a limit point of $S$, so I ...
1
vote
2answers
52 views

Prove $\sup(A)=\sup(B)$

Let there be $A,B\in \mathbb{R}$ non-empty, let assume that $\forall a\in A \exists b \in B$ such that $a\leq b$ and $\forall b\in B \exists a \in A$ such that $b\leq a$. Show that $A$ is ...
0
votes
3answers
41 views

Prove that a number $u$ is $\sup S$ given certain properties.

Problem Let $S$ be a nonempty subset of $\mathbb{R}$, and let $u$ be a number with the following properties: for each positive integer $n$, the number $u - \frac{1}{n}$ is not an upper bound of ...
0
votes
1answer
10 views

Supremums of sequences

Let $x_n$ and $y_n$ be two sequences of real numbers. Assume that $y_n$ is bounded above and that $x_n$$<$$y_n$ for all n$\in$$\mathbb{N}$. (a) Prove that $x_n$ is also bounded above. (b) Prove ...
0
votes
2answers
20 views

Proving supremums of sequences

If $l$=sup($x_n$), what is sup($kx_n$) where k$\in$$\mathbb{R}^{+}$? Prove your conjecture. I have that sup($kx_n$)=$kl$. I can prove that it is an upper bound of $kx_n$, but I'm having trouble ...
0
votes
1answer
24 views

condition for a supremum

let $A\subseteq \mathbb{R}$ be a non-empty set and $s\in \mathbb{R}$ and upper bound of $A$. So $s$ is the supremum of $A$ $\iff$ $\forall \epsilon>0$ there is $x\in A$ so $s-\epsilon<x\leq s$. ...
1
vote
2answers
54 views

Find the infimum of the set $S=\left\{\frac{1}{m}-\frac{1}{n} \, : m,n \in \mathbb{N^+}\right\}$

I need to find the infimum of the set $$S=\left\{\frac{1}{m}-\frac{1}{n} \, : m,n \in \mathbb{N^+}\right\}$$ and formally prove that it is indeed the infimum of $S$. From intuition, I know the $\inf ...
0
votes
1answer
33 views

Supremum/Infimum of the set $(x^2)-x<1$ where $x$ belongs to the rationals?

$$A = \{x \in \mathbb Q : (x^2)-x<1\} = \{x \in \mathbb Q : (1-\sqrt{5})/2 < x < (1+\sqrt{5})/2\}$$ Then it says therefore the supremum $= (1+\sqrt{5})/2$ and infimum $= (1-\sqrt{5})/2$. But ...
0
votes
1answer
45 views

lim inf sup integral

I have a general question about integrals of sequence of functions. Suppose $f_n \rightarrow f$ pointwise. Can I automatically say that $\lim_{n\rightarrow \infty} f_n = \lim_{n\rightarrow \infty}inf ...
1
vote
2answers
50 views

Finding $\sup$ of $\{\sqrt{n} - \lfloor\sqrt{n}\rfloor | n\in\mathbb{N}\}$

The question is to find $\sup$ and $\inf$ of $B=\{\sqrt{n} - \lfloor\sqrt{n}\rfloor | n\in\mathbb{N}\}$ where $\lfloor x \rfloor$ is defined as the largest integer that is smaller than $x$. That ...
0
votes
2answers
26 views

Prove that if $a$ is a real number with $a >2$, then there is an $n$ is an element of natural number such that $2+1/\sqrt{n}<a$

Prove that if $a$ is a real number with $a >2$, then there is an $n$ is an element of natural number such that $2+1/\sqrt{n}<a$ The goal is to show $\inf\{2+1/\sqrt{n} : ...
1
vote
1answer
35 views

Supremum vs. Maximum in the definition of the Lp norm [duplicate]

The $L_p$ norm $||A||_p$ is defined as $$\sup_{x \neq 0} \frac{||Ax||_p}{||x||_p} = \max_{||x||_p = 1} ||Ax||_p \tag{1}$$ I'm not quite getting why the LHS uses $\sup$ but the RHS uses $\max$. I ...
1
vote
1answer
40 views

Finding $\sup$ and $\inf$ of $\frac{n^5}{2^n}$ where $n$ is natural number

I'm trying to find $\sup A, \inf A$ where $$A=\{a_n=\frac{n^5}{2^n}:n\in\Bbb{N}\}, 1\not\in\Bbb{N}$$ For $n=1$ we have $a_1 = \frac{1}{2}$, $\lim_{x\rightarrow +\infty} \frac{n^5}{2^n}=0$ and after ...
2
votes
1answer
15 views

How to prove a certain inequality for a complex valued function using its derivative?

I have the following question: Let $n\in \mathbb{N}\setminus \{0\}$ be an arbitrary, but fixed natural number, let $f:\mathbb{R}^{+}\rightarrow \mathbb{C}:\ t\mapsto n^{-b} - t^{-b}$, when $b\in ...
0
votes
1answer
34 views

Is A∩B has Maximum?

Consider $A=(0,1)∪\{2\}$ and $B=(0,1)∪\{3\}$. Both sets have maxima, $2$ and $3$ respectively. But $A∩B=(0,1)$ which has no maximum. How can we say that $A∩B=(0,1)$ does not have any maximum. ...
1
vote
2answers
90 views

Prove that, for $s$ is upper bound of A, $s = \sup A$ iff , if $r < s$, so there exists $x \in A$ such that $r < x \leq s$.

Could someone verify my proof? Definition: Suppose $s \in \mathbb{R}$ and upper bounded $A \subset \mathbb{R}$. For any $x \in A$, we have $x \leq s$. For any $v$ such that $x \leq v$ for any $x$, we ...
2
votes
3answers
46 views

$\sup$ and $\inf$ of this set

These are exercises from my textbook, and I am not sure if the solutions are correct or not. Given a set $B = \{\frac{n}{2n+1} : n \in \mathbb{N} \}$ Find the $\sup$ and $\inf$ of $B$, and maxima ...
0
votes
0answers
13 views

Determening global sup and inf points in a function

I have function: f(x)=(x+0.4)^2-cos(1+ctg(0.5x)) and I need to find global inf and sup in interval of xe[-1;1]. I tried to do the following, but with no avail: Determin domain of a function: ...
0
votes
1answer
58 views

Does the empty set have a minimum if it is a subset of R?

True or False: As a subset of $\bf{R}$: $\emptyset$ has a minimum. What is the difference between a supremum and the maximum? are they used interchangeably?
1
vote
1answer
36 views

Real analysis, boundaries question

Let $A$ be a non empty set of real numbers and let $f,g$ be functions defined on $A$ such that $f$ is bounded above on $A$ and $g$ is bounded below on $A$ . If $f(x)\le g(x) \;\forall\; x \in A$. ...
1
vote
3answers
56 views

Proof or counterexample : Supremum and infimum

If $($An$)_{n \in N}$ are sets such that each $A_n$ has a supremum and $∩_{n \in N}$$A_n$ $\neq$ $\emptyset$ , then $∩_{n \in N}$$A_n$ has a supremum. How to Prove This.
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votes
1answer
36 views

Supremum And infimum True or false

If the sets A and B have maxima and A ∩ B $\neq$ 0 , then A ∩ B has a maximum. Is this statement is true or false ? How to do these kind of problems
2
votes
1answer
52 views

Are all finitely distributive and join-complete lattices infinitely distributive?

The infinite distributive law on a join-complete lattice $L$ is as follows: $\displaystyle a \wedge\left( \bigvee_{b \in B} b \right) = \bigvee_{b \in B}(a \wedge b) $ for all $a \in L$ and $B ...