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

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1answer
44 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
36 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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0answers
44 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
39 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 $n$ ...
2
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0answers
36 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
21 views

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 ...
2
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2answers
58 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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0answers
22 views

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
15 views

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
18 views

“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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0answers
45 views

Proving that $\lim_{n \to\infty} \frac{X_n}{n} = 0$

If ${X_n}$ are nonnegative random variables such that $\sup_{n\ge1} E(X^a_n) \lt \infty$ where a $\gt$ 1 is a constant. Prove that $$\lim_{n \to\infty} \frac{X_n}{n} = 0$$ Now my question is, what ...
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1answer
28 views

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
34 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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1answer
18 views

Show supA(∪B)=supB given sup(A∪B)=u [closed]

Suppose that sup(A∪B)=u and that there is an ε>0 so that a < u-ε for all a∈A. Show that supA(∪B)=supB. Any help would be appreciated on solving this!
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1answer
34 views

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
13 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
41 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
29 views

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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votes
2answers
31 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
48 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 ...
1
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2answers
51 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 ...
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1answer
26 views

Findin the $\sup$ and $\inf$ of defined sets

Yet again struggling. Find $\sup A$ and $\inf A$ where $A$ is the set defined by: (a) $A=\{x∈\mathbb{Q}:x^{2} −x<1\}$ (b) $A=\{x∈\mathbb{R}:x^3 −x\le6\}$ My answers: (a) $\sup A= ?\quad\inf ...
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0answers
35 views

If $u\notin E$, then the supremum of $E\cup\{u\}$ is $\sup\{\sup E, u\}$

Let $S$ be an ordered set and $S \supset E $. Let $\alpha = \sup E \in E$. If $u \notin E$, then we have $ \sup ( E \cup \{u\} ) = \sup\{ \alpha, u \} $. Try: I know this result follows easy by ...
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3answers
124 views

Show $\inf_f\int_0^1|f'(x)-f(x)|dx=1/e$ for continuously differentiable functions with $f(0)=0$, $f(1)=1$.

Let $C$ be the class of all real-valued continuously differentiable functions $f$ on the interval $[0,1]$ with $f(0)=0$ and $f(1)=1$. How to show that $$\inf_{f\in ...
1
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1answer
27 views

An element $u$ is an upper bound of $E$ if and only if $t>u$ implies $t\notin E$

Let $S$ be an ordered field and $S \supset E\neq \varnothing$. Then, the following are equivalent: $u \in S$ is an upper bound of $E$. $t \in S$ and $t > u$ implies $t \notin E $. My Try: ...
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3answers
46 views

Infimum of $\left\{\frac{n}{n^2+1}\:\:;\:n\:\in \mathbb N\right\}$ with a proof

Consider $A=\left\{\frac{n}{n^2+1}\:\:;\:n\:\in \:N\right\}$. I need to find and prove $\inf(A)$. So I know that I need to prove that for every $\epsilon > 0$ exists some $a\:\in A$ such that ...
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2answers
39 views

infimum and supremum of subsets question

Let $B \subseteq \mathbb{R}_{+}$ such that B is non-empty. consider $B^{-1} = \left \{b^{-1} : b\in B \right \}$. Show that if $B^{-1}$ is unbounded from above, then $\inf\left(B\right)=0$ How can i ...
2
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1answer
42 views

The supermum of E

Let $f\ [0,1]\longrightarrow [0,1]$ be increasing function. let: $$E=\{x\in [0,1] \mid f(x)\geq x \} $$ Show that $E$ has a supermum $b$ and that $f(b)= b$. we have $x\leq 1$ since $f$ is ...
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2answers
61 views

The supremum of set $A=\left\{ \left(\frac{m+n+1}{m+n}\right)^{m+n};\ m,n\in\mathbb{N}^* \right\}$.

Let $A=\left\{ \left(\dfrac{m+n+1}{m+n}\right)^{m+n};\ m,n\in\mathbb{N}^* \right\}$. Calculate the supremum of $A$ ($\sup A$). I tired To find the first few elements of $A$ ...
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2answers
35 views

Finding the supremum of a set

Consider the sets A and B defined by $$A=\{ \frac{x}{x^2+7}: x \in \Bbb R\} \ \ and \ \ b=\{ \frac{x}{x^2+7}: x \in \Bbb N\} $$ What are the values of sup A and sup B? For the first part of the ...
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2answers
55 views

For every $n$ there exists $k_n \in \mathbb{N}$ such that $a+k_n/2^n$ is an upper bound while $a+(k_n-1)/2^n$ is not

Let $ \mathcal{P} \subset \mathbb{R}$,\ $\mathcal{P}\neq \emptyset $ and let $b$ be an upper bound of $\mathcal{P}$. Let $a \in \mathcal{P}$ and let $n\in \mathbb{N}^*$ Show that : ...
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2answers
72 views

show $\exists\ m\in\mathbb{N} \text{ such that: } \quad a+\dfrac{m}{2^n}\geq b$

Let $ \mathcal{P} \subset \mathbb{R}$, $\ \mathcal{P}\neq \emptyset $ et let $b$ an upper bound of $\mathcal{P}$ Let $a \in \mathcal{P}$ and let $n\in \mathbb{N}^*$ Show that : ...
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0answers
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How can one take supremum and infinimum of a sum in rieman integral to define integrability?

We define a function is integrable if sup{L(f,p)}=inf{U(f,p)} where sup is supremum and inf is infinimum p is a partition function. But how can we take supremum of a number as L(f,p) is defined as ...
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2answers
42 views

How to prove the infimum of this subset

consider $A\:=\:\left\{\frac{n-m}{n+m}\:;\:m<n\:\:n,m\:\in \mathbb{N}\right\}$ How to prove that $1$ is the supremum and $0$ is the infimum? I go to the definition of infimum, and see that by ...
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3answers
69 views

Find $\bigcap_{n=1}^\infty(0,1/n)=\emptyset$ [closed]

I`ve tried this and is it true or completely not? Then how can I fix it? Proof: too wrong so I get it off
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1answer
17 views

Certain property of supremum

Given real numbers $x_{ni}, n\in \mathbb{N}, i\in I$, does it hold that $$\sup \bigg\{ \sum_{n\in\mathbb{N}}x_{ni}|i\in I\bigg\}=\sum_{n\in \mathbb{N}}\sup\bigg\{x_{ni}|i\in I \bigg\}?$$ Thanks in ...
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3answers
35 views

Is supremum part of the set or it is the bigest element out of it?

"If $\sup A$ is in $A$, then is $\sup A$ called also Maximum: $\max A$." So that means that $\sup A$ can be outside the set $A$? And lastly the upper barrier(or bound, not sure) is from $A$, ...
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1answer
41 views

Prove the infimum and supremum of the positive rational numbers

I am having this set: $$ X= \mathbb{Q^+} = \{x \in \mathbb{R} \ \ |x \in \mathbb{Q} \ \text{and} \ x>0 \} $$ How can I prove that $\inf X= 0$ and there is no supremum ? (I think there is no ...
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3answers
28 views

Prove Infimum and Supremum

I am having this set: $$ X= (-1.1) = \{x \in \mathbb{R} \ \ |-1<x<1 \} $$ How can I prove that $\inf X= -1$ and $\sup X=1$ ? (I think there is no maximum and no minimum in X?)
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0answers
18 views

Equation with CDF

My problem is to confirm following lemma (or give a counterexample to prove that it's wrong) Let $G$ be a CDF of unknown density (we don't know if it's continuous or not). Is it true that for $x$ ...
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4answers
45 views

Prove $\sup(f^2(x))\le(\sup(f(x))^2$

Prove $\sup(f^2(x))\le(\sup(f(x))^2$ I am assuming this is true, I could find no counter examples. $f(x)\le\sup(f(x))$ $f^2(x)\le(\sup(f(x)))^2$ when $f(x)\ge0$ $\sup(f^2(x))\le(\sup(f(x))^2$ when ...
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1answer
16 views

Proving an equality of supremums related to the rate function

Given $\sigma > 0$, $$\sup_{\lambda \in \mathbb{R}} \left\lbrace \sigma \lambda\left(\frac{x-m}{\sigma}\right) - \ln \int e^{\lambda \sigma t} \, \mu(dt) \right\rbrace = \sup_{\lambda \in ...
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0answers
93 views

supremum is the only positive root of $z^n=a^m$

I'm trying to define $a^x$ with x rational.For $r=\frac{m}{n}$ for $ m,n \in \mathbb{Z}$ and $a>1$ is real. We define $S_{r}(a)= \left \{ x \in \mathbb{R} | 0\leq x^n\leq a^m \right \} $. Now we ...
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3answers
41 views

prove $(\sup{|f(x)|})^2\geq(\sup{f(x)})^2$

prove $(\sup{|f(x)|})^2\geq(\sup{f(x)})^2$. $|f(x)|\geq f(x)$ $\sup{|f(x)|}\geq\sup{f(x)}$ $(\sup{|f(x)|})^2\geq(\sup{f(x)})^2$ when $\sup{f(x)}\geq0$ ... I don't know how to prove the other half ...
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3answers
50 views

“sup” in an equation

I am currently reading through JC Lagarias' "The $3x+1$ Problem and its Generalizations" and have come across some notation reading : $$\sup_{K \ge 0} T^{(K)}(N)$$ Now I assume that this means ...
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0answers
19 views

Finding infimum of a set

$\varnothing\neq A\subseteq\mathbb{R}$ is defined as follows: $\forall a \in A \exists b \in A$ such that $b\le \frac{a}{2}+1$ Show that $\inf A \le 2$ Maybe the question is simple, still i do not ...
0
votes
1answer
56 views

If $f$ is continuous on $[a,b]$ and $F(x) = \sup f([a,x])$. Prove that $F$ is continuous on $[a,b]$ .

Exercise: Suppose that $f$ is continuous on $[a,b]$ and that $F(x) = \sup f([a,x])$. Prove that $F$ is continuous on $[a,b]$ . Attempt of proof: Suppose that $f$ is continuous on $[a,b]$ and that ...
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0answers
38 views

A question related to Integral and supremum

Let $f\in L_{p}([0,1])$ and 1-periodic on $R^{1}.$ Suppose $[a,c]\subset [0,1].$ Are the following quantities equal? $$ \underset{|h|\leq \delta_{1}}{\sup}\int_{a}^{b}|f(x+h)-f(x)|^{p}dx+ ...
0
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3answers
28 views

Subset of a set problem with least upper bound

Suppose that $S$ and $T$ are (non-empty) sets in $\mathbb{R}$, such that $T$ is bounded from above and $S\subseteq T$. Prove that $S$ is bounded from above and that $$ \sup(S) \leq \sup(T). $$ I ...
2
votes
1answer
55 views

Am I right that this statement is false?

For every nonempty set $A$ of real numbers having an upper bound, and for every $d \in \mathbb{R}$, we examine the statement: $$[(\sup A = d, d \notin A) \implies (\exists_{N \in ...