For questions concerning the definition and properties of limit superior and limit inferior.

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2
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2answers
35 views

Show That $\limsup_{x \to x_0} f(x) \ge \liminf_{x \to x_0} f(x)$

My question concerns proving an inequality between two extreme limits, namely: $$\limsup_{x \to x_0} f(x) \ge \liminf_{x \to x_0} f(x)$$ Using the following defintions: Let $f: E \to \mathbb{R}$ be a ...
1
vote
0answers
18 views

Is the limit superior defined on every real-valued function with a finite range?

I know the limit superior isn't always defined on function in general, but intuitively, I don't see how it could be undefined on a function with a finite range. However, I didn't find any sources ...
2
votes
0answers
16 views

$\limsup$ of Brownian Motion Time Integral

The following are well-known: $\limsup_{t\rightarrow \infty} \frac{B(t)}{t} = 0$ $\limsup_{t\rightarrow \infty} \frac{B(t)}{\sqrt t} = \infty$ $\limsup_{t\rightarrow \infty} \frac{B(t)}{\sqrt ...
1
vote
0answers
51 views

What is bigger, $\liminf $ or $\inf ( \liminf )$

Let $E$ be a linear normed vector space, $f$ a functional $f : E \rightarrow \mathbb{R} $ and $ S = \{g \in E : \|g\| = 1\} \subset E$. Fix $x \in E$. I need to prove that $$\liminf_{g \in S, \ ...
0
votes
1answer
49 views

proof that $\limsup a_n=\sup\{a_n,a_{n+1},…\}$

How can I prove that $\limsup a_n= \sup\{a_n,a_{n+1},...\}$? I also need to prove: for two sequences $a_n>0$ and $b_n \ge 0$, then $\limsup(a_n b_n) \le \limsup(a_n) \limsup(b_n)$. I thought ...
1
vote
3answers
33 views

Do we need to have a subsequence such that $\lim_{k\to\infty}\left\|x_{n_k}\right\|=\liminf_{n\to\infty}\left\|x_n\right\|$?

Let $(X,\left\|\;\cdot\;\right\|)$ be a normed space and $(x_n)_{n\in\mathbb{N}}\subseteq X$. Can we prove that there is a subsequence ...
0
votes
0answers
12 views

swap limsup with a function

What are the conditions that must hold for a sequence of real numbers $\{a_n\}$ and a real valued function $f$ so that the following relation holds: $$\limsup_{n\rightarrow\infty} f(a_n)= f( ...
-1
votes
1answer
26 views

(analysis) sequence a(n) is bounbed and sequence b(n) converges. Show that a(n)≤b(n) (∀ n ∈ ℕ) ⇒lim supa(n)≤lim b(n) [closed]

sequence a(n) is bounbed and sequence b(n) converges. Show that a(n)≤b(n) (∀ n ∈ ℕ) ⇒lim sup a(n)≤lim b(n) Since a(n) is bounded, a(n) has a convergent subsequence. let a1'(n) be a subsequence of ...
-1
votes
2answers
35 views

Let $ f:I \to \mathbb{R} , I=(0,1) $ be uniformly continuous. Then exists $ \lim_{n\to\infty} f(\frac{1}{n}) $

True. Since $f$ is continuous (because all uniformly continuous function is continuous), we can assume: $$ f\left(\lim_{n\to\infty} \frac{1}{n}\right) $$ Since $ \lim_{n\to\infty} \frac{1}{n} $ is ...
0
votes
1answer
27 views

Find $\liminf\limits_{n\to\infty} (x_{n})$ and $\limsup\limits_{n\to\infty}(x_{n})$ for a sequence $x_{n}=1-nsin\frac{n\pi}{4}$

Subsequence $a_{n_{1}}=sin\frac{n\pi}{4}$ is bounded ($[-1,1]$), and a subsequence $a_{n_{2}}=n$ is bounded below. We can find cluster points for the first subsequence $C_{1}=\{-1,1\}$. For the second ...
1
vote
1answer
24 views

My attempt at finding $\underline{\lim_{n \to \infty}}A_n$ and $\overline{\lim_{n \to \infty}}A_n$ $A_n=B$ for $n$ even; $A_n=C$ for $n$ odd.

My attempt at finding $$a.) \underline{\lim_{n \to \infty}}A_n$$ and $$b.) \overline{\lim_{n \to \infty}}A_n$$ $A_n=B$ for $n$ even; $A_n=C$ for $n$ odd. $a.)=\bigcup_{n=1}^{\infty} ...
0
votes
1answer
46 views

Asymptotic sums and liminf

Given an arithmetic function $f(n)>0$ with $$\liminf \frac{g(n)}{f(n)}=C$$ for a certain constant $C$ and another function $g(n)>0$, in the study of the asymptotic bound for $$ \sum_{n\leq x} ...
0
votes
2answers
29 views

If $a_n \geq b_n$, $b_n \leq |c_n|$ with $c_n \to 0$, does this imply $\liminf a_n \geq 0?$

Let $a_n \geq b_n$ where $b_n \leq |c_n|$ with $c_n \to 0$. Does this imply that $$\liminf a_n \geq 0?$$ These are all real-valued sequences. I don't think it is enough to conclude.
2
votes
0answers
13 views

When working with the limites inferior and superior, how do you procede?

There are typical cases of various complexity where notions are defined or solvable by liminf or limsup. At the same time, from my perception, they are touched only superficially during the course of ...
0
votes
2answers
31 views

Yes or No: The following $\infty\triangleq\sup{\mathbb{R}}$ and $-\infty\triangleq \inf\mathbb{R}$ holds

Can someone verify whether $\infty\triangleq\sup{\mathbb{R}}$ and $-\infty\triangleq \inf\mathbb{R}$ is mathematically rigorous?
0
votes
1answer
11 views

Limit superior and inferior of a sequence that satisfies the asymptotic formula $\sum\limits_{n\leq x} a_n \sim x $

Suppose $\{a_n\}$ is a sequence such that $\sum\limits_{n\leq x} a_n \sim x $. I have to show that: $$ \liminf a_n\leq 1 \leq \limsup a_n$$ I've no idea on how to approach this, in all honesty.
0
votes
0answers
30 views

Filter,dual ideal,definition of $\liminf_I \lambda$

We have this http://shelah.logic.at/files/506.pdf definition of $\lim \inf_I \bar{\lambda}$ in 1.1(3): for $I$ a filter on $\kappa$ let $I^+=2^\kappa \setminus I$.$$\lim \inf_I ...
0
votes
1answer
42 views

Prove a limsup and liminf inequality.

Let $(a_n)_{n\in\mathbb{N}}$ be a bounded sequence and let $B_n = \frac{1}{n} \sum_{i=1}^n a_i$ for each $n \in \mathbb{N}$. Prove that $\liminf a_n \le \liminf B_n \le \limsup B_n \le \limsup a_n$. ...
0
votes
0answers
33 views

References for limit superior and limit inferior of functions

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}\cup\{\pm \infty\}$. I would like to find books for the following notions and their properties: 1) $\limsup_{x\rightarrow x_0}f(x), \liminf_{x\rightarrow ...
0
votes
1answer
17 views

Limit inferior implication

I'm having trouble with the following statement. If $M=\liminf_{t\uparrow b}|x(t)|$ then $\exists$ a sequence $t_n\rightarrow b$ such that |$x(t_n)| \leq M+1$ I take it that with the sequence ...
1
vote
2answers
159 views

Computing a limit of sequence

After applying l'Hopital rule twice, one sees that $$ \lim_{n\to \infty} n a ~e^{-an} =0 \qquad \qquad (a\in [0,1]) . $$ I would like to ask if someone can prove it using different way? Bests.
3
votes
2answers
31 views

Is $\limsup_{z\to z_0}f(z)=\limsup_{k\to\infty}f(z_k)$?

Let $\Omega\in\Bbb C$ open, $f:\Omega\to\Bbb R$ a generic function. Let $(z_k)_k\subset\Omega$ s.t. $\lim_{k}z_k=:z_0\in\Omega$. The question is the following: is true that $$ \limsup_{z\to ...
0
votes
1answer
33 views

Proof of Fatou-Lebesgue Theorem

Good evening everyone, how can I prove the following inequality? Let $f,f_n\in L(X,\mu)$ on a measure space with $0\leq f_n(x)\leq f(x)$. If $f,f_n$ are $\mu$-almost everywhere in $X$, then is ...
0
votes
1answer
34 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 ...
1
vote
3answers
29 views

Finding two sequences with a limsup value

Find two sequences ⟨ X n ⟩ and ⟨ Y n ⟩ such that limsup(Xn + Yn )=E=Euler's constant and limsup Xn + limsup Yn =Pi=π . I couldn't ...
0
votes
0answers
18 views

Liminf Brownian Motion question

For this assignment I'm working on, I was able to prove that: $$\limsup_{t \rightarrow \infty} \frac{B_t}{\sqrt t} = \infty$$ where $B_t$ is a Brownian Motion. I'd like to be able to prove: ...
1
vote
2answers
48 views

Isn't the statement of the Fatou's lemma somewhat problematic?

My lecture notes define $\int f := \int f^+ - \int f^-$ provided both $\int f^{\pm}$ are finite. And then the Fatou's lemma is stated in the following way: Let $f_n$ be a sequence of integrable ...
4
votes
1answer
56 views

If $\limsup_n\sqrt[n]{a_n}=\frac{1}{r}$, then $\limsup_n\sqrt[n]{(n+1)a_{n+1}}=?$

If $\limsup_n\sqrt[n]{|a_n|}=\frac{1}{r}$, then $\limsup_n\sqrt[n]{|(n+1)a_{n+1}|}=?$ Can I separate the product of the second limit as ...
1
vote
3answers
64 views

If $\lim_{n\to \infty}a_n = a\in \mathbb{R}$ . Prove that $\limsup_{n\to \infty}a_n x_n=a\limsup_{n\to \infty}x_n$ .

Note: $x_n$ is a sequence which is not necessarily convergent. The following was my attempt. Since $\lim_{n\to \infty}a_n=a$ then $\limsup_{n\to \infty}a_n=a$ . Also ...
4
votes
0answers
37 views

Multiples of some set has density

Nathanson gives a proof in Elementary Methods in Number Theory (Theorem 7.14) that, if a set $S$ of positive integers has $$ \sum_{s\in S}\frac1s<+\infty $$ then the set of positive multiples of ...
1
vote
2answers
69 views

Excerise 12.2 from Ross Elementary Analysis

I'm having a little bit of difficulty proving this question: Prove that $ \limsup_{n->\infty} |S_n| = 0 ~~$iff$ ~\lim_{n->\infty}S_n = 0. $ What I have so far: $ (\Leftarrow)$ $ $Suppose $ ...
1
vote
1answer
27 views

Every sequence of sets have coverging subsequence

Is that true, that every sequence of sets have coverging subsequence? We say that sequence of sets $A_1, A_2, A_3, ...$ coverging iff ${\limsup}_{n \to \infty} A_n = {\liminf}_{n \to \infty} A_n$
1
vote
1answer
34 views

What is the limsup of the following sequence of sets?

Problem: Let $A_n=\{\frac{m}{n}:m\in\mathbb{Z}\}$. Find $\limsup_{n\rightarrow\infty}A_n$ and $\liminf_{n\rightarrow\infty}A_n$ Attempted: It is clear that limsup should be $\mathbb{Q}$. I can show ...
1
vote
1answer
47 views

Give an example to show that the inequalities are strict inequalities

Give an example to show that the following three inequalities $$\liminf_{n \to \infty} (a_n) +\liminf_{n \to \infty} (b_n)\le\liminf_{n \to \infty} (a_n+b_n)\le\limsup_{n \to \infty} (a_n+b_n) \le ...
0
votes
1answer
41 views

Doubt on Tail events and Kolmogorov Zero-One Law

In this wiki article on the law of the iterated logarithm, one states that, given $M>0$, the event $$A=\left\{\limsup_{n \to \infty} \frac{S_n}{\sqrt{n}} > M\right\}$$ has probability $0$ or ...
2
votes
0answers
52 views

how to show $\lim \sup_{n \to \infty} \frac{\phi(n+1)}{\phi(n)}= \infty$ and $\lim \inf_{n \to \infty} \frac{\phi(n+1)}{\phi(n)}= 0$?

Let $\phi (n)$ be the Euler's totient function . Thenhow to prove that the set $\{\dfrac{\phi(n+1)}{\phi(n)}: n \in \mathbb Z^+\}$ is unbounded above that is how to show $\lim \sup_{n \to \infty} ...
1
vote
1answer
35 views

Prove that $\limsup_{n \to \infty} (a_n+b_n)= a + \limsup_{n \to \infty} b_n$

I have to prove the following: Given that lim $a_n$ exists and that lim $a_n$ = $a\in \mathbb{R}$ prove that: $$\limsup_{n \to \infty} (a_n+b_n)= a + \limsup_{n \to \infty} b_n$$ I proved one ...
0
votes
1answer
23 views

limit superior and inferior of measurable sets

I am trying to prove the following: Let $(A_n)_{n \geq 1}$ be a sequence of measurable sets, then (i) $|\lim \inf_{n \to \infty} A_n| \leq \lim \inf_{n \to \infty} |A_n|$ (ii) If there is $n \in ...
2
votes
2answers
73 views

how that if $P(\lim \sup A_n) = 1$ then, $P(\bigcup_{n=1}^\infty A_n)=1$

Question: Let $\{A_n\}$ be a sequence of independent events in a probability space $(\Omega, F, P)$ show that if $P(\lim \sup A_n) = 1$ then, $P(\bigcup_{n=1}^\infty A_n)=1$ I tried solving this ...
1
vote
1answer
26 views

Proving a subsequence from lim inf

I'm trying to solve this problem, Let $a_n$ be a sequence such that lim inf$ |a_n| = 0 $. Prove that there is a subsequence $a_{n_k}$ such that $\sum a_{n_k}$ converges. So far, I tried to say we ...
1
vote
1answer
26 views

Prove absolute convergence for a summation

I need help with this problem. I've been staring at the page blankly tyring to think of ways to solve it. Any hints/solutions would be greatly appreciated. If $ \displaystyle \lim_{n \to ...
3
votes
1answer
43 views

Prove equality in liminf of set

I need help in proving $\chi_{\liminf A_n} = \liminf \chi_{A_n}$ Where $\liminf A_n=\bigcup \limits_{n=1}^{\infty}\bigcap \limits_{k=n}^{\infty}A_k$ and $\chi_{A}$ is the characteristic ...
0
votes
0answers
25 views

A quick question on the limsup of sets and logical connectives

I know we can interpret $x\in\limsup{A_n}$ iff $x$ is in infinitely many of the $A_n$'s, but am confused as to which logical connective we need to use to describe when $x\in\limsup{A_n}$, when we ...
0
votes
2answers
47 views

liminf inequality in measure spaces

Let $(X;\mathscr{M},\mu)$ be a measure space and $\{E_j\}_{j=1}^\infty\subset \mathscr{M}$. Show that $$\mu(\liminf E_j)\leq \liminf \mu(E_j)$$ and, if $\mu\left(\bigcup_{j=1}^\infty ...
0
votes
1answer
78 views

Find $\lim \sup A_n$ and $\lim \inf A_n$?

Question: Let $\Omega = R^2. A_n$ is the interior of a circle with center at $\{\frac{(-1)^n}{n},0 \} $ at radius 1. Find $\lim \sup A_n$ and $\lim \inf A_n$? My answer is the following; Let $\Omega ...
3
votes
2answers
37 views

Proving that $\mathcal{L}(x_n+y_n) \subset {\cal L}(x_n)+{\cal L}(y_n)$.

Let $(x_n)$ and $(y_n)$ be two bounded real sequences. Let: $${\cal L}(x_n) =\{ L \in [-\infty,+\infty] \mid L \text{ is the limit of some subsequence }(x_{n_k}) \},$$ and the same for ${\cal L}(y_n)$ ...
0
votes
1answer
23 views

Need help with: $\lim \sup x_n = \sup{z_k}$ where $z_k$ is a set of limit points

This question stems from another one, but presents my concern in a more specific way. There is a theorem saying that: $\lim \sup x_n = \sup{z_k}$ where $z_k$ is a set of limit points for the ...
1
vote
1answer
19 views

$\lim \sup x_n = \sup{z_k}$ where ${z_k}$ is a set of limit points for the sequence ${x_n}$

There is a theorem saying that: $\lim \sup x_n = \sup{z_k}$ where ${z_k}$ is a set of limit points for the sequence ${x_n}$. All sets and sequences are real. Limit point for a sequence is a point ...
-1
votes
1answer
30 views

Weakest assumption for $\lim \sup x_n \in \mathbb R$

What is the weakest assumption to be satisfied so that $\lim \sup x_n \in \mathbb R$. The same question for $\lim \inf x_n$. Note that $\mathbb R$ does not include $\pm \infty$. Should $x_n$ ...
0
votes
0answers
13 views

If expectation is bounded away from 0, is then the probabilty of being positive also positive?

Consider a discrete-time stochastic process $(Y_t)_{t \geq 0}$, where we know that $$\liminf_{t \rightarrow \infty} \mathbb{E}[Y_t] >0.$$ Does this imply that $$\liminf_{t \rightarrow \infty} ...