Questions tagged [supremum-and-infimum]

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

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Let $2, 1+\frac{1}{2}, 3, 1+\frac{1}{3}, 4, 1+\frac {1}{4},\dots$ be a sequence. Does $a_n$ converge/diverge? Is there a $\sup$ or $\inf$?

Let $2,1+\frac{1}{2},3,1+\frac{1}{3},4,1+\frac {1}{4},...$ be a sequence then which of the statements is true? $a_n$ coverges to a finitie limit or diverges to infinity. $\limsup \limits_{n \to \...
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Exercise A.14(b) in "Supplement for Measure, Integration & Real Analysis" by Sheldon Axler.

I am reading "Supplement for Measure, Integration & Real Analysis" by Sheldon Axler. I have no idea about Exercise A.14(b). Please tell me a solution to Exercise A.14(b). Exercise A.14: ...
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2 votes
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Conjecture about areas of circular segment and polygon with equal perimeter sharing a side

I was playing around with shapes and have formed a conjecture. Length of the red circular arc $=$ total length of the $n$ green line segments Conjecture: $$\sup{\left(\frac{\text{Area}_1}{\text{Area}...
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1 vote
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Show that the expectation of supremum is bounded

Assume that $X_1, X_2, \dots$ are independent real random variables with means $\mu_1, \mu_2, \dots$ and variances $\sigma_1^2, \sigma_2^2, \dots$ such that $\displaystyle\sum_{k=1}^\infty \dfrac{\...
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  • 460
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Moving supremum

Let $E \subset \mathbb C$ be a compact set, and $t > 0, a \in \mathbb C$. We define function: $$f_E(z) = \sup\{(\frac{|p(z)|}{\|p\|_E})^{\frac{1}{\deg p}}:p \in P(\mathbb C^N), \deg p \ge 1, p_E \...
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2 votes
2 answers
60 views

Find $\sup$ and $\inf$ of $x\sin(\frac{1}{x})$ on $(0,\infty)$.

Find $\sup$ and $\inf$ of $x\sin(\frac{1}{x})$ on $(0,\infty)$. I found an example. Example: Let $f(x)=x\sin(\frac{1}{x})$. We are interested in its behaviour for $x\geq1$. $f'(x)=\sin(\frac{1}{x})-\...
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Finding the supremum, infimum, and bounds of $f(x) = x^2$ for $x \le a$ and $f(x) = a + 2$ for $x > a$ in the interval $(-a-1,a+1)$, where $a > -1$.

My current progress; $x^2$ is increasing, so it will always have a minimum of $0$. Furthermore, if $a > -1$, then $a + 2 > 1$ and $-a -1 < 0$. Therefore, $f(x) \ge 0$. I tried to split $f$ ...
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Set of measures on the simplex of matrices that take support on rank 1 matrices has a minimal element

Let $M=\{ A\in\mathbb R^{n\times m} : \forall i,j~A_{i,j}\geq 0,\sum_{i,j} A_{i,j} = 1 \}$ be the simplex of $n\times m$ matrices, it was shown in a previous question that the set $R$ of rank $1$ ...
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3 votes
3 answers
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Prove that the set of reciprocals of natural numbers has no positive infimum.

Prove that for each $x>0$, there is an $ n \in \mathbb N $ such that $\frac{1}{n}$ $<$ $x$, WITHOUT using the Archimedean theorem. Its simple enough with the theorem, but without the theorem I ...
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  • 165
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3 answers
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ways to calculate norm of the Bounded Linear Functional.

$\Lambda$ be a linear functional on $C([0,1])$ defined by $$ \Lambda(f) = \int_0^1 xf(x)dx \;\;\; \text{ for } f \in C([0,1]). $$ and use $\|f\|_{sup} = \sup_{x \in [0,1]} |f(x)| $ for $f \in C([0,1])$...
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Supremum of bounded family of functions

Let $X$ be a metric space and $\mathcal{F}$ family of functions from $X$ to $\mathbb{R}$. If $\mathcal{F}$ is bounded above in $X$, then $\sup_{f\in \mathcal{F}} f(x)<\infty$. So far, I read in ...
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Supremum with respect to an object

In my Real Analysis class I've often seen the Upper Riemann Sum (in the definition of Riemann integrability) defined as: $$\sup_P\{U(f,P)\}$$ This seems to mean the supremum of all possible upper sums ...
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Let $A \subset \mathbb{R}$ and $c \in \mathbb{R}$. If $c \geq 0$, then $\sup{cA} = c\sup{A}$ [duplicate]

Let $A \subset \mathbb{R}$ and $c \in \mathbb{R}$. If $c \geq 0$, then $\sup{cA} = c\sup{A}$. How to prove this theorem? I am thinking of deriving to the inequalities $\sup{cA} \geq c\sup{A}$ and $\...
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supremum of integral over probability measures

Let $f,g,h: (\mathbb{S}^n)^3 \to \mathbb{R}$ be some bounded and continuous functions such that: $f(\theta,\varphi,\psi)=g(\theta,\varphi,\psi)+h(\theta,\varphi,\psi)$ for all $(\theta,\varphi,\psi)\...
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1 answer
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supremum Proof by contradiction .

(The second part of the solution is confusing me) The Question: A is a bounded subset of ℝ B = {b|b = 2a + 3, a ∈ A} Prove that Sup(B) = 2Sup(A) + 3 Part 1 of solution: We get 2a + 3 ≤ 2Sup(A) + 3 So ...
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5 votes
2 answers
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How to prove that $\sup(A-B) = \sup(A) - \inf(B)$?

How to prove that $\sup(A-B) = \sup(A) - \inf(B)$? My attempt: Let $c \in A-B$ and define $c= a-b$, where $a \in A$ and $b \in B$. Then $a-b \leq \sup(A) - \inf(B)$. Hence, $\sup(A-B) \leq \sup(A) - \...
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When is $\sup (f+g)= \sup f + \sup g$? [closed]

What are some sufficient conditions on two functions $f,g$ to have $$\sup (f+g)= \sup f + \sup g?$$
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Show $\inf\{t\ge0:x(t-)\in B\text{ or }x(t)\in B\}\le T$ iff $x(T)\in B=\bigcap_nB_n$ or $\forall n:\exists s\in[0,T):x(s)\in B_n$

Let $(E,d)$ be a metric space, $x:[0,\infty)\to E$ be càdlàg (right-continuous with left-limits), $x(0-):=x(0)$, $x(t-):=\lim_{s\to t-}x(s)$ for $t>0$, $B\subseteq E$ be nonempty and $$\tau:=\inf\...
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1 vote
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Prove $P\big(\lim (\inf A_n)\big) \le\big( \lim \inf P(A_n)\big)$

Let we have $(A_n: n\ge 1)$ a succession of events in a probability space. I have to prove that: $P\big(\lim (\inf A_n)\big) \le \lim \big(\inf P(A_n)\big)$ $\lim \big(\sup P(A_n)\big) \le P(\lim \...
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1 vote
1 answer
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Why is product of infimum and supremum of a commuting pair of elements in a lattice-ordered group equal to the product of the elements?

Context: Self-study. Seth Warner's Modern Algebra (1965), question $15.11$ gives: If $(G, \circ \preccurlyeq)$ is a lattice-ordered group and if $x$ and $y$ are commuting elements of $G$, then $$\sup ...
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  • 4,048
0 votes
1 answer
34 views

Supremum of $\alpha^2 z^\gamma/(\alpha + z)^2$ for $\gamma \in (0, 2)$

I want to calculate $\sup\limits_{z} \left( \dfrac{\alpha}{\alpha + z} \right)^2 z^{\gamma}$ for $\gamma \in (0, 2)$. I know that the solution is $z = \dfrac{\gamma \alpha}{2 - \gamma}$ but I don't ...
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Uniform convergence through Supremum [duplicate]

I came up with a solution to a simple problem that I was working on but it seemed too simplistic so I doubt that my solution is correct which is why I want some help. $f_n(x)$ is a sequence of complex ...
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  • 496
3 votes
1 answer
119 views

Does this infimum converge to $+\infty$?

Consider a sequence of $\{ \phi^n\} \subset \mathcal{C}^2(\mathbb{R})$ such that \begin{align} &\left|\frac{\partial}{\partial x} \phi^{n}(x)\right| \leq M \\ &\lim _{n \rightarrow \infty} \...
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0 votes
1 answer
24 views

About approximation property of supremum [closed]

Let $A$ be a nonempty upper bounded set. So I know that for all $\varepsilon>0$, $\sup A<a+\varepsilon$, where $a$ is an arbitrary element of $A$. But since $x<y+\varepsilon$ (for all $\...
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3 votes
0 answers
114 views

Can a norm on polynomials be "almost multiplicative", even for large degrees?

Definition: A norm on a real algebra is called almost multiplicative if there are positive constants $L$ and $U$ such that, for all $f$ and $g$ in the algebra, $$L\lVert f\rVert\cdot\lVert g\rVert\;\...
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1 vote
1 answer
68 views

Is infimum on nonnegative function $>0$ in an example

Let $f,g,h$ be bounded funtions on the domain $[0,T]$ with $0<f<1$ $g, h >0$ $\inf_{t\in[0,T]}((1+f(t))g(t)-h(t))>0 \enspace (1)$ $\inf_{t\in[0,T]}(h(t)-(1-f(t))g(t))>0 \enspace (2)$ ...
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0 votes
1 answer
48 views

Upper bounds, infimum and supremum

I have a question about bounds and others. So, question is that: $D$ is the set $D = \bigl\{ \frac{(-1)^n n}{n+1}\colon n \in \Bbb{N} \bigr\}$. How to find $\inf(D)$ and $\sup(D)$ and show they do not ...
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3 votes
1 answer
180 views

How we can we prove that for any $b > a, x > 0$, $\frac{2}{\pi} (1-\frac{a}{b})<\sup|\frac{\sin(ax)}{ax} - \frac{\sin(bx)}{bx}|<4(1-\frac{a}{b})$

I want to prove this for any $x > 0$ and $b > a > 0$: $$ \frac{2}{\pi} (1-\frac{a}{b})<\sup|\frac{\sin(ax)}{ax} - \frac{\sin(bx)}{bx}|<4(1-\frac{a}{b}) $$ I tried the derivation to find ...
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0 votes
1 answer
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Is it always true that the maximum of a closed bounded set, $[a,b]$, is also the supremum

This seems true and I've seen some tangential proofs in my real analysis class but I wasn't 100% sure. I think this is the case because there can be no bound smaller than the maximum.
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1 vote
1 answer
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If a sequence of random variable is bounded in probability(Op(1)), then will its supremum be finite with probability 1(P (supn|Xn|<infinity)=1)?

I am practicing for my large sample theory course, and stumbled upon this problem and got stuck. New to the subject so kind of stumped how to proceed.
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0 votes
1 answer
31 views

Let B={$\frac{m^2-n}{m^2+n^2}: n,m \in \mathbb{N}, m>n$}. Prove $supA=1$ and $infA=\frac{1}{2}$

I am preparing for my exam and need help with the following task: Let B={$\frac{m^2-n}{m^2+n^2}: n,m \in \mathbb{N}, m>n$}. Prove $supA=1$ and $infA=\frac{1}{2}$. Well at first, I thought that we ...
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0 answers
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Useful properties for integration

Proposition (1). If $f:[a,b]\to\mathbb{R},g:[a,b]\to\mathbb{R}$ are bounded functions and $P:=\{x_0,x_1,x_2,...,x_n\}$ is a partion of $[a,b]$ then $$m_i(f)+m_i(g)\leq m_i(f+g),$$ where for every ...
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-1 votes
4 answers
81 views

Infimum of $\left\{\left|\sqrt{m} - \sqrt{n}\right|\;: \;m,n \in\mathbb{N},\; m≠n\right\}$ [closed]

In my homework assignment I encountered this problem: Find the infimum of the set $A=\left\{\left|\sqrt{m} - \sqrt{n}\right|\;: \;m,n \in\mathbb{N},\; m≠n\right\}$ How do i even start to find the ...
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2 votes
0 answers
12 views

Inequality searching for an upper-bound

Suppose that there exists a constant $C$ such that the following relation holds for all $G$: \begin{equation*} \vert T(F)-T(G) \vert \le C \sup_y \vert F(y)-G(y) \vert \end{equation*} Suppose that ...
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2 votes
0 answers
14 views

$\underset{\omega}{\max}\; \omega^{\top}\cdot F(\omega') \leq \underset{\omega}{\max}\; \omega^{\top}\cdot F(\omega),\; \forall \omega'?$

I have a question: Is it true that $$ \underset{\omega}{\max}\; \omega^{\top}\cdot F(\omega') \leq \underset{\omega}{\max}\; \omega^{\top}\cdot F(\omega),\; \forall \omega' $$ where the elements of ...
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Exercise 4, Section 24 of Munkres’ Topology

Let $X$ be an ordered set in the order topology. Show that if $X$ is connected, then $X$ is a linear continuum. My attempt: Approach(1): first we show second property of linear continuum. Assume ...
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1 vote
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How do I prove that the following set is limited?

I have this set: $M=\left\{\frac{1}{n^{2}}-\frac{2}{m}:n,m\in\mathbb{Z}\setminus \left\{0 \right\}\right\}$ How do I prove it's bounded? Additionally, how can I decide the infimum and supremum? I ...
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2 votes
1 answer
90 views

How can you show this convergence for the supremum?

Fix $\delta>0$ and consider the function $\Phi_\delta \colon \mathbb{R}^N \times \mathbb{R}^N \to \mathbb{R}$ defined by $$\Phi_\delta(x,y)=w(x)-v(y)-\delta |x|^2-\delta |y|^2$$ where $w,v \in \...
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1 vote
1 answer
75 views

If $a_0<\dots<a_n<b_n<\dots<b_0$ for any $n\in\Bbb N$ then $\sup( a_n)\le\inf(b_n)$?

Let be $(a_n)_{n\in\Bbb N}$ and $(b_n)_{n\in\Bbb N}$ and increasing and a decreasing sequence of reals numbers such that $$ a_0<\dots <a_n<b_n<\dots<b_0 $$ for any $n\in\Bbb N$. So by ...
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6 votes
1 answer
124 views

Oscillation of the function at each point

I would like to find the oscillation of the function $f:\Bbb R^2\to\Bbb R,$ $$f(x,y)=\begin{cases}\sin\left(\frac1x\right)+\sin\left(\frac1y\right),&x,y\ne 0\\0, x,=0\text{ or } y=0.\end{cases}$$ ...
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0 votes
1 answer
19 views

The infimum over all partitions is the same as the infimum over all partitions including a fixed partition $Q$

Take an interval $[a,b]$ on the real line and a bounded function $f:[a,b] \to \mathbb R$. For a partition $P =\{ a = t_0, t_1, \dots, t_n = b\}$ of said interval we define the Upper Darboux Sum of $f$ ...
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0 answers
11 views

Arg max for extenced-reals-valued functions

Let $X$ be a susbset of a topological space and let $f:X \mapsto \overline{\mathbb{R}}$ be a function having as codomain the extended real line. Usually, we say that $f$ has a maximum whenever the ...
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0 answers
22 views

Finding index of sequence that satisfies given estimation

How do I find the minimal $ n\in \mathbb{N} $ of sequence $ x_{n}=\frac{4+\sqrt{3n}}{3+\sqrt{3n}} $? Such that the estimation $ |x_{n}-1|<\frac{1}{10}, n\geqslant N $ holds? My idea was to find ...
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0 votes
1 answer
32 views

Prove the sequence $\frac{nz}{n+1} + \frac{3}{n}$ converges uniformly to $z$ on $|z| \leq R|$ for all $R$.

My attempt: Given any $\epsilon > 0$, \begin{align} \sup_{|z| \leq R}\, \biggl\lvert \frac{nz}{n+1} +\frac{3}{n} - z \biggr\rvert &= \sup_{|z| \leq R}\, \biggl\lvert \frac{nz-nz-z}{n+1} + \...
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0 votes
1 answer
71 views

Proof that {$\frac{1}{b_n}$} is bounded and $\limsup\limits_{n\rightarrow\infty}$ $\frac{1}{b_n}$=$\frac{1}{\liminf\limits_{n\rightarrow\infty}b_n}$

I need help/suggestions for improvements for a task, that I try to solve as a preparation for my exam. Let {${b_n}$}$_{n\in\mathbb{N}}$ be a bounded sequence with $b_n \neq 0$ $\forall$ n $\in\mathbb{...
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0 votes
1 answer
103 views

What is the essential supremum?

I've encountered something called the "essential supremum" while working with $L^{p}$ spaces (in particular, for $p=\infty$). I tried looking it up on the internet but all the definitions ...
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1 vote
0 answers
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$M_{n+1}=\{\frac{1}{2}(a+b):a,b\in M_n\}\cup\{\frac {1}{2}ab:a,b\in M_n\}, n\in\mathbb {N_0}$. Find bounds of ${M_n}$ independent from $n$

I am preparing for my analysis exam by trying to solve old exam tasks from my professor, since he often uses his tasks from his earlier exams. I need help with this task: Let $M_0=\{1,2\}$ and $$M_{n+...
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0 votes
0 answers
46 views

Almost sure convergence for absolute value of random variables and inequalities and supremums

I read this article about almost sure inequality and the supremum from Almost sure inequality and the supremum Let $X_n$ be a sequence of random variables, if $\left|X_n \right| \leq Y$ almost surely ...
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0 votes
1 answer
52 views

Why is d' a metric in this case?

If $X$ is a compact space and $Y$ a metric topological space with metric $d$, then $d'(\alpha,\beta) = sup_{x \in X} d(\alpha (x),\beta (x))$ is a metric on $TOP(X,Y)$ $TOP(X,Y)$ denotes the set of ...
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  • 286
0 votes
1 answer
22 views

Proving that the limit of a sequence is the maximum element given that the sequence is bigger than the supremeum.

The exact problem goes like: Let some $A$ be a bounded and non-empty set. There is a sequence ${x}$ which converges to some point $x'$ in $A$ and I have to prove that $x'$ is the max($A$) given that ...
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