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).
2,606
questions
7
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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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0
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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
1
answer
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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:
$$\sup{\left(\frac{\text{Area}_1}{\text{Area}...
- 1,192
1
vote
0
answers
33
views
Show that the expectation of supremum is bounded
Assume that $X_1, X_2, \dots$ are independent real random variables with means $\mu_1, \mu_2, \dots$ and variances $\sigma_1^2, \sigma_2^2, \dots$ such that $\displaystyle\sum_{k=1}^\infty \dfrac{\...
- 460
0
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0
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25
views
Moving supremum
Let $E \subset \mathbb C$ be a compact set, and $t > 0, a \in \mathbb C$. We define function:
$$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 \...
- 239
2
votes
2
answers
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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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votes
0
answers
12
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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 $f$ ...
- 253
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0
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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 $1$ ...
- 3,965
3
votes
3
answers
44
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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 ...
- 165
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3
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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 $f \in C([0,1])$...
- 521
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votes
1
answer
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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 ...
- 263
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29
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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
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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 ...
- 67
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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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1
answer
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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 $$\tau:=\inf\...
- 12.8k
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2
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Prove $P\big(\lim (\inf A_n)\big) \le\big( \lim \inf P(A_n)\big)$
Let we have $(A_n: n\ge 1)$ a succession of events in a probability space. I have to prove that:
$P\big(\lim (\inf A_n)\big) \le \lim \big(\inf P(A_n)\big)$
$\lim \big(\sup P(A_n)\big) \le P(\lim \...
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vote
1
answer
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Why is product of infimum and supremum of a commuting pair of elements in a lattice-ordered group equal to the product of the elements?
Context: Self-study.
Seth Warner's Modern Algebra (1965), question $15.11$ gives:
If $(G, \circ \preccurlyeq)$ is a lattice-ordered group and if $x$ and $y$ are commuting elements of $G$, then $$\sup ...
- 4,048
0
votes
1
answer
34
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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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0
answers
25
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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 ...
- 496
3
votes
1
answer
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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} \...
- 71
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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votes
0
answers
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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,
$$L\lVert f\rVert\cdot\lVert g\rVert\;\...
- 4,150
1
vote
1
answer
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views
Is infimum on nonnegative function $>0$ in an example
Let $f,g,h$ be bounded funtions on the domain $[0,T]$ with
$0<f<1$
$g, h >0$
$\inf_{t\in[0,T]}((1+f(t))g(t)-h(t))>0 \enspace (1)$
$\inf_{t\in[0,T]}(h(t)-(1-f(t))g(t))>0 \enspace (2)$
...
user711386
0
votes
1
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Upper bounds, infimum and supremum
I have a question about bounds and others.
So, question is that:
$D$ is the set $D = \bigl\{ \frac{(-1)^n n}{n+1}\colon n \in \Bbb{N} \bigr\}$.
How to find $\inf(D)$ and $\sup(D)$ and show they do not ...
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votes
1
answer
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How we can we prove that for any $b > a, x > 0$, $\frac{2}{\pi} (1-\frac{a}{b})<\sup|\frac{\sin(ax)}{ax} - \frac{\sin(bx)}{bx}|<4(1-\frac{a}{b})$
I want to prove this for any $x > 0$ and $b > a > 0$:
$$
\frac{2}{\pi} (1-\frac{a}{b})<\sup|\frac{\sin(ax)}{ax} - \frac{\sin(bx)}{bx}|<4(1-\frac{a}{b})
$$
I tried the derivation to find ...
0
votes
1
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Is it always true that the maximum of a closed bounded set, $[a,b]$, is also the supremum
This seems true and I've seen some tangential proofs in my real analysis class but I wasn't 100% sure. I think this is the case because there can be no bound smaller than the maximum.
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If a sequence of random variable is bounded in probability(Op(1)), then will its supremum be finite with probability 1(P (supn|Xn|<infinity)=1)?
I am practicing for my large sample theory course, and stumbled upon this problem and got stuck. New to the subject so kind of stumped how to proceed.
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Let B={$\frac{m^2-n}{m^2+n^2}: n,m \in \mathbb{N}, m>n$}. Prove $supA=1$ and $infA=\frac{1}{2}$
I am preparing for my exam and need help with the following task:
Let B={$\frac{m^2-n}{m^2+n^2}: n,m \in \mathbb{N}, m>n$}. Prove $supA=1$ and $infA=\frac{1}{2}$.
Well at first, I thought that we ...
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Useful properties for integration
Proposition (1). If $f:[a,b]\to\mathbb{R},g:[a,b]\to\mathbb{R}$ are bounded functions and $P:=\{x_0,x_1,x_2,...,x_n\}$ is a partion of $[a,b]$ then
$$m_i(f)+m_i(g)\leq m_i(f+g),$$
where for every ...
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4
answers
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Infimum of $\left\{\left|\sqrt{m} - \sqrt{n}\right|\;: \;m,n \in\mathbb{N},\; m≠n\right\}$ [closed]
In my homework assignment I encountered this problem:
Find the infimum of the set $A=\left\{\left|\sqrt{m} - \sqrt{n}\right|\;: \;m,n \in\mathbb{N},\; m≠n\right\}$
How do i even start to find the ...
2
votes
0
answers
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Inequality searching for an upper-bound
Suppose that there exists a constant $C$ such that the following relation
holds for all $G$:
\begin{equation*}
\vert T(F)-T(G) \vert \le C \sup_y \vert F(y)-G(y) \vert
\end{equation*}
Suppose that ...
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0
answers
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$\underset{\omega}{\max}\; \omega^{\top}\cdot F(\omega') \leq \underset{\omega}{\max}\; \omega^{\top}\cdot F(\omega),\; \forall \omega'?$
I have a question:
Is it true that
$$
\underset{\omega}{\max}\; \omega^{\top}\cdot F(\omega') \leq \underset{\omega}{\max}\; \omega^{\top}\cdot F(\omega),\; \forall \omega'
$$
where the elements of ...
- 23
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votes
0
answers
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Exercise 4, Section 24 of Munkres’ Topology
Let $X$ be an ordered set in the order topology. Show that if $X$ is connected, then $X$ is a linear continuum.
My attempt:
Approach(1): first we show second property of linear continuum. Assume ...
- 1,393
1
vote
0
answers
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How do I prove that the following set is limited?
I have this set:
$M=\left\{\frac{1}{n^{2}}-\frac{2}{m}:n,m\in\mathbb{Z}\setminus \left\{0 \right\}\right\}$
How do I prove it's bounded? Additionally, how can I decide the infimum and supremum? I ...
2
votes
1
answer
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views
How can you show this convergence for the supremum?
Fix $\delta>0$ and consider the function $\Phi_\delta \colon \mathbb{R}^N \times \mathbb{R}^N \to \mathbb{R}$ defined by
$$\Phi_\delta(x,y)=w(x)-v(y)-\delta |x|^2-\delta |y|^2$$
where $w,v \in \...
- 71
1
vote
1
answer
75
views
If $a_0<\dots<a_n<b_n<\dots<b_0$ for any $n\in\Bbb N$ then $\sup( a_n)\le\inf(b_n)$?
Let be $(a_n)_{n\in\Bbb N}$ and $(b_n)_{n\in\Bbb N}$ and increasing and a decreasing sequence of reals numbers such that
$$
a_0<\dots <a_n<b_n<\dots<b_0
$$
for any $n\in\Bbb N$. So by ...
- 3,432
6
votes
1
answer
124
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Oscillation of the function at each point
I would like to find the oscillation of the function $f:\Bbb R^2\to\Bbb R,$ $$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}$$
...
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$ ...
- 2,012
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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 ...
- 1,452
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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 ...
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} + \...
- 75
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1
answer
71
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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{...
- 238
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votes
1
answer
103
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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 ...
- 11
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vote
0
answers
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views
$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+...
- 238
0
votes
0
answers
46
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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 ...
- 133
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votes
1
answer
52
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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 ...
- 286
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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 ...
