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

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Proving that $\sup f'\left( \left( 0,\infty \right) \right)=0$ under a certain set of conditions.

Let $f$ be a twice differentiable function on $\left( 0,\infty \right)$ s.t. $f''(x)>0$ for all $x\in \left( 0,\infty \right)$. Prove, that if the following conditions are satisfied: ...
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0answers
22 views

Relationship between supremum of the partial derivative and the derivative of the supremum

To fully set up the problem: I have a function $$F : \mathbb R^n \times \mathbb R^+ \to \mathbb R$$ such that $F$ is positive and for any fixed $t^* > 0$, $$F(x,t^*) \in H^\infty(\mathbb R^n);$$ ...
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1answer
39 views

How do I prove this statement about the operator norm?

I stumbled across this equation in a paper, which may seem obvious, but I'm wondering if someone can explain why this is true? By definition of an operator norm, $$\left[(D^*D)^{-1} - ...
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0answers
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Properties of infimum and supremum

I have a convex function to optimize: $Sup_{\alpha:1^T\alpha=0} inf_y [C\sum_{t=1}^T(V(y_t,r_t)+\alpha_ty_t)-\frac{1}{2}\alpha^TK\alpha]$ where V is the loss function, and I would like to derive the ...
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23 views

I want to know that the supremum function continuous [on hold]

Let $g(y)=\sup_{x\in[0,y]}f(x)$ for $y\ge0$. I want to know that the function $g(y)$ is continuous on $[0,\infty)$. (here we suppose $f(x)$ is continuous on $[0,\infty)$)
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21 views

Let $Q \subset (m,n)$ be a subset which open and closed, show that $ Q = (m,n)$ or $ Q = \emptyset$

Consider the metric space $((m,n),d)$ where $(m,n) \subset \mathbb{R}$ and $d(x,y) = |x-y|$ Let $Q \subset (m,n)$ be a subset which open and closed, show that$ Q = (m,n)$ or $ Q = \emptyset$ there ...
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2answers
28 views

Is the support function always unique for a convex set?

Given an arbitrary set $A ⊂ \mathbb{R}^n$ , the support function associated with the set $A$ $ σ_A : \mathbb{R}^n \to \mathbb{R} ∪ \{+\infty\}$ is defined as $\sigma_A(x):= \sup_{z \in A} \langle ...
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1answer
27 views

Support function of a set is convex

Given an arbitrary set $A ⊂ \mathbb{R}^n$ , the support function associated with the set $A$ $ σ_A : \mathbb{R}^n \to \mathbb{R} ∪ \{+\infty\}$ is defined as $\sigma_A(x):= \sup_{z \in A} \langle ...
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1answer
29 views

The set of separating hyperplanes is a convex cone

Let $A, B ⊂ \mathbb{R}^n$ be two nonempty sets such that $A ∩ B = ∅$. $H(A, B) := \{(w, d) ∈ \mathbb{R}^{n+1} : \sup_{x\in A} \langle w,x\rangle ≤ d ≤ \inf_{y \in B} \langle w, y\rangle \}$ I can't ...
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2answers
62 views

Is $\lim\sup=\sup\lim$?

Assume $(a_n(x))_{n=1}^{\infty}$ is a bounded sequence in $\mathbb R$, when $x$ is $\in\mathbb R$ and is relevant to the sequence in some way that doesn't really interest us in my question. Assume ...
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1answer
37 views

Min, Max, Infimum, Supremum

I am trying to understand the notion of minimum, maximum, infimum and supremum. Can you please comment on these solutions for the below examples? Minimum , Maximum, Infimum, Supremum : a.$(0,1)$ ...
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0answers
46 views

Difference between lower sums and lower integral

Why is it true that the lower sums of f with respect to some partition is less than the lower integral (which is the supremum of the lower sums) I think what I'm confused about is the difference ...
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1answer
17 views

Show $\alpha$ is a limit ordinal $\leftrightarrow \alpha \neq 0$ and $\cup \alpha = \alpha$

$\alpha$ is a limit ordinal $\leftrightarrow \alpha \neq 0$ and $\cup \alpha = \alpha$ Sorry if this question has been asked already but I couldn't find it on this site. I assume by definition ...
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1answer
39 views

infimum and supremum notation

I have stumbled across this blob of text when reading my textbook, and would like to know how to interpret it more intuitively. I understand the definitions of inf and sup, however not so much what ...
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2answers
58 views

Can anyone give me a hint on how to prove that subadditivty of the limit supremum of two sequences? Everything I can find says “it's trivial.”

I'm working through some analysis textbooks on my own, so I don't want the full answer. I'm only looking for a hint on this problem. I'm trying to prove the subadditivity of the limit supremum, i.e. ...
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Why is the limit supremum an increasing sequence? Is my textbook incorrect?

My textbook defines the limit supremum and limit infimum of a sequence $(a_i)$ as $$ b_k = \sup\{a_i : i \ge k\} \\ c_k = \inf\{a_i : i \ge k\} $$ The book also states that $b_k \le b_{k+1}$ and ...
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2answers
51 views

Does maximum or supremum of an infinite set exit?

If I have a set: $S = \{ k_1, k_2,...,k_n,... : k_i < \infty \}$, which means that $S$ include infinite many elements, but each element is a finite real number. In this case does $\max(S)$ or ...
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0answers
21 views

Does sequence of quantiles $x_{p_n}$ converge to endpoint $x_F$ of cdf F?

Let $X$ be a RV with continuous cdf $F$. Let $p_n$ be a sequence in $(0,1)$ with $\lim_{n \to \infty}p_n = 0$ and $x_{p_n}=\inf \{y \in \mathbb{R}:F(y) \ge 1-p_n\}$ the $p_n$-quantile of F. Define by ...
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2answers
50 views

(Dis)prove that $\sup(A \cap B) = \min\{\sup A, \sup B\}$

Just beginning real analysis so I'm having some trouble with disproving this statement: $$\sup(A \cap B) = \min\{\sup A, \sup B\}$$ Initially it asks whether it's true or false and to provide a ...
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1answer
28 views

Compute the supremum $\sup_{x \leq t \leq x+1}f(t)$

I have to consider the functions $f$ and $g$ defined on $\mathbb{R}$ by $$f(x)=(2x-1)^2 \quad \text{ and }\quad g(x)=\sup_{x \leq t \leq x+1}f(t)$$ The goal is to compute explicitely $g(x)$. ...
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1answer
32 views

Proof of theorem in infimum and supremum

The first line is the statement that I want to prove. Let A and B be bounded non-empty subsets of R. Can someone please tell me does my proof (especially second last line) of this question valid or ...
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1answer
27 views

Proof on infimum and supremum

Sorry for the poor photo quality. Can someone please tell me does my proof of this question valid or not?
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60 views

Find the infimum (with proof) of the set $ X = \left \{ x \in \mathbb{R} : x^{2} < 3 \right \} $ [closed]

I know the answer is $-\sqrt{3}$ but I don't know how to give a formal proof. Thanks in advance!
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2answers
75 views

Show that sets of real numbers $A, B$ are adjacent iff $\sup A = \inf B$

If $A,B \subset \mathbb{R}$ satisfy : $$\begin{cases}\forall\ a \in A,\ \forall\ b \in B,\ a \le b \cr \forall\ \epsilon > 0,\ \exists\ a \in A,\ \exists\ b \in B \text{ such that }\quad ...
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1answer
43 views

Showing that $\inf\left|x-p\right|$ is continuous

Let $X\subset R$ be non-empty. $f :\Bbb{R}\to\Bbb{R}$ is defined by $$f(p)=\inf_{x\in X}\left|x-p\right|$$ for every $p\in\Bbb{R}$. How do I show that $f$ is continuous? I tried using the reverse ...
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2answers
75 views

Properties of the supremum and infimum.

Consider the set $S=\{x\in\mathbb{R}:x^{2}<2\}$. Is it true that $\sup{(S)}$ must also be a real number? If so, what would be the proof strategy?? Is this also true for the infimum of $S$?
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1answer
56 views

Take $S$ to be a set of ordinals, show $\cup S=\sup(S)$

Ok so firstly I can see how this makes sense. If you have say: $S=\{0,1,2\}$ with $0 = \emptyset, 1 = \{0\} = \{\emptyset\}, 2 = \{0,1\} = \{\emptyset,\{\emptyset\}\}$. $$\cup S = ...
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1answer
64 views

For $B=\{|x-y|: x,y\in A\}$, show that $\sup B = \sup A - \inf A$ and find $\inf B$

I'm trying to solve this exercise: Let $A$ be a non-empty set $\mathbb R$-bounded. Let $B=\{|x-y|: x,y\in A\}$. Prove that $B$ has a least upper-bound and a greatest lower-bound. ...
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1answer
31 views

$infA\leq SupA$

Prove that if there is $A\subseteq \mathbb{R}$ not empty and bounded so $infA\leq SupA$ and infA=SupA iff A={One element}. By definition every bounded set in $\mathbb{R}$ has Inf and Sup, therefore ...
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2answers
42 views

Does $\displaystyle\liminf_{n\to \infty} -a_{n}= -\displaystyle\limsup_{n\to \infty}a_{n}$?

Let $(a_{n})$ be a bounded sequence. How to prove $$\displaystyle\liminf_{n\to \infty} -a_{n}= -\displaystyle\limsup_{n\to \infty}a_{n}$$ I don't how formally prove this..can someone guide me? tnx!
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1answer
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2 question about supremum of subset and a sequence that converge to it.

Let $A$ be a bounded subset of $\mathbb{R}$. 1. Show that there exists a sequence $a_n$ of elements of $A$ such that $\lim _{ }\left(a_n\right)\:=\:sup\left(A\right)$ 2. Show that we can build a ...
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1answer
64 views

Finding points that satisfy $f(a) = \sup f(x)$

Choose positive real numbers $\alpha_1,\ldots,\alpha_n$, $n$ such that $\sum_{i=1}^n \alpha_i = 1$ and let $$f: [0,\infty)^n \to \mathbb R$$ $$x=(x_1, \ldots, x_n) \mapsto x_1^{\alpha_1} \cdots ...
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0answers
43 views

Property of infimum

if $f(x)$ is a continuous, bounded, nondecreasing function defined in $]0,\infty[$, so the limit for $x \rightarrow 0$ exists and it is equal to the infimum. Assume the infimum is $0$. In my course ...
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1answer
72 views

how to add supremums

I need to prove that $$\sup(S)+\sup(T)=\sup(S+T)$$ I don't understand what $\sup(S+T)$ means, can you show me examples for sets $S$ and $T$ for which this equation holds?
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2answers
44 views

The limit of upper bounds is also an upper bound

Question We have a set E which is a subset of the real numbers. There is a sequence ${x_n}$ such that $\{x_n\} \subseteq E$. Suppose there is another sequence $\{y_n\}$ such that the limit as $n$ ...
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0answers
44 views

Norm for a set of vectors

Let V be a normed vector space (real or complex valued) with norm $\|\cdot\|_V$. For any nonempty and bounded subset $A \subseteq V$ one can define $\|A\|$ via $$\|A\|:=\sup\{|x|:x\in A\}$$ I ...
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2answers
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If $\sup A = 5$ and $B = \left\{ 3a \mid a \in A \right\}$ then $\sup B = 15$

Prove that if $A \subset \mathbb{R}$, $\sup A = 5$, and $B = \left\{ 3a \mid a \in A \right\}$, then $\sup = 15$. I tried to do contradiction by assuming the hypothesis and that there is a number ...
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103 views

Supremum of a sine integral

Let $M_T=\int\limits_{0}^{T}\frac{\sin(t)}{t}dt$ be a sine integral. Why is $2\displaystyle\sup_{T}M_T < \infty$?
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If the supremum is finite, then the value is attained.

In a linear programming proof we have the: $\sup\{c^Tx: Ax \le b\}$ This supremem can be $\infty$, or defined as $-\infty$, if there are no vectors x such that $Ax \le b$. But it is stated that ...
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1answer
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On the Limiting Behavior of Sequences: Limit Supremums

I'm having difficulty understanding the following inequality as presented in Kenneth A Ross's Elementary Analysis: The Theory of Calculus $(2^{nd} Edition)$. For a sequence $(s_n) \in \mathbb R$, ...
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2answers
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“Greatest lower bound function”

If $f $ is a function continuous at $c, h $ is positive and $m$ is a function defined as $ m(h)=\inf \{ f(x): x \in [c,c+h] \}$ , how can I prove that the limit of $ m $ as $ h $ approaches $ 0 $ ...
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1answer
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How to find supremum and infimum of set [closed]

How to find supremum and infimum of set $\{\frac{nk}{1 + 2n + 3k} : n,k \in \mathbb{N}\}$ when $0 \notin \mathbb{N}$?
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1answer
50 views

How to find infimum and supremum

I have to find the infimum and supremum of the set $A = \left\{\frac{n + k^2}{2^n + k^2 + 1} : n,k \in \mathbb{N}\right\}$. We assume $0 \notin \mathbb{N}$. $\inf A = 0$ because $\lim_{n \to ...
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Let A⊆R be a set. Prove that A is bounded if and only if there is some M∈R such that M>0 and that |x|≤M for all x∈A.

I proved the above problem as follows but received feedback that I'm not certain I understand. Can someone help me determine where I went wrong in the proof? Let A⊆R be a set. Suppose M∈R where ...
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1answer
18 views

question on least upper bound principle.

Show that the Least Upper Bound Principle does not hold for the rational numbers. This is to say, show that there is a subset S of Q which is bounded above but such that there is no L ∈ Q which is a ...
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3answers
43 views

Calculate $\sup \,\left\{\frac{n}{n+1}:n \in\mathbb{N}\right\}$

Calculate $\sup \,\left\{\frac{n}{n+1}:n \in\mathbb{N}\right\}$ Can anyone help me with this? I am very confused with this question. Thank you.
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0answers
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Equivalence of weak $L^p$ norms

I'm kind of new to the subject of weak $L^p$ spaces. The definition of the (quasi-)norm in weak $L^p$ ($p\in(0; \infty)\,$) over $\sigma$-finite measure space $(X, \mu)$ I use is $||f||_{L^{p, ...
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2answers
32 views

Prove least upper bound for piecewise inequality

Say $S = \left\{ x \in \mathbb{Q} \mid 0 < x < 1 \textrm{ or } 3 < x < 4 \textrm{ or } 6 < x < 8 \right\}$. I want to show that the least upper bound of this set is $8$, but I don't ...
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1answer
58 views

Question about supremum $\implies$ infimum

For all the proofs I've seen that if a set has the least upper bound property, then that set also has the greatest lower bound property, they assert something like if a set has lower bounds, then the ...
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2answers
54 views

Maximum and minimum function on an interval

Let $I := [a,b]$, where $a<b$. Suppose that $f$ is continuous and $1-1$ on $I$. Let $m$ denote the minimum value of $f$ on $I$ and let $M$ denote the maximum value of $f$ on $I$. (a) Carefully ...