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

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Prove the following properties of sequence

Define $$L = \limsup_{k \rightarrow \infty}a_k =\inf_j(\sup_{k\geq j}a_k).$$ Prove that if $(a_k)$ and $(b_k)$ are sequence of real numbers then $$\limsup(a_k + b_k) \leq \limsup a_k + \limsup b_k.$$ ...
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1answer
18 views

Wheeden-Zygmund exercise

Define $\limsup_{k \rightarrow \infty}a_k$ and $\liminf_{k \rightarrow \infty}a_k$ by $$\limsup_{k \rightarrow \infty}a_k = \lim_{j\rightarrow \infty}b_j = \inf_{j}\{\sup_{k\geq j}a_k \} $$ ...
2
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1answer
25 views

$ 1+\mathsf{d}^\star(B) \le \mathsf{d}^\star(A\cup B)+\mathsf{d}^\star(B\cup C) $ with $A\cup B\cup C=\bf{N}^+$

For each subset of positive integers $X\subseteq \bf N^+$ define the upper asymptotic density as $$ \mathsf{d}^\star(X)=\limsup_{n\to \infty} \frac{|X\cap [1,n]|}{n}. $$ Problem: Let $A,B,C$ be a ...
0
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1answer
29 views

A question on convergent series of positive terms

Let $\sum a_n$ be a convergent series of positive terms ; then we know $\lim \inf (na_n)=0$ ; can we derive from here that if $\{a_n\}$ is decreasing , then $\lim (na_n)=0$ ?
2
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1answer
66 views

The proof of theorem 3.19 from baby Rudin

If $s_n\leqslant t_n$ for $n\geqslant N$, where $N$ is fixed, then $$\liminf_{n\to \infty} s_n\leqslant \liminf_{n\to \infty} t_n$$ $$\limsup_{n\to \infty} s_n\leqslant \limsup_{n\to \infty} t_n$$ ...
5
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2answers
74 views

Does $\lim_{n\to \infty}\sigma_n=0\implies \lim_{n\to \infty} a_n=0$?

Let $\{a_n\}$ be a sequence of positive real numbers such that $\lim_{n\to \infty} \sigma_n=0$ where $\sigma_n=\left(\sum_{k=1}^n a_k\right)/n$. Does $\lim_{n} \sigma_n=0\implies \lim_n a_n=0$? ...
0
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1answer
25 views

Is $\exists p\in\mathbb{N}:\frac{1}{\left| n^p \sin(n/2) \right|}$ is bounded for $n \in \mathbb{N}$?

It's clear that $\frac{1}{\left| n^p \sin(n/2) \right|}$ is not bounded where $n \in \mathbb{R}$ because $\frac{1}{\left| n^p \sin(n/2) \right|} \to \infty$ as $n \to 2k\pi\ (k \in \mathbb{N}).$ ...
0
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1answer
22 views

$\limsup_n a_n + \limsup_n b_n \le \limsup_n c_n + \limsup_n d_n$ if $a_n+b_n=c_n+d_n$ and $a_n$ maximum

I already posed some questions on inqualities between superior limits of real sequences, so here there is another one: Let $(a_n)_{n\ge 1}, (b_n)_{n\ge 1}, (c_n)_{n\ge 1}$, and $(d_n)_{n\ge 1}$ be ...
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1answer
35 views

Some basic questions about definitions of sup an inf

I am looking to see if anyone could help to conform to me if my intuition is correct, or if not please explain how I can understand it better. For example, if I wanted to say find the sup, inf of the ...
0
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1answer
19 views

$\limsup_n a_n+b_n+c_n+\limsup_n b_n \le \limsup_n a_n+b_n+\limsup_n b_n+c_n$?

Let $(a_n)_{n\ge 1},(b_n)_{n\ge 1},(c_n)_{n\ge 1}$ be sequences of reals in $[0,1]$. Is it true that $$ \limsup_n (a_n+b_n+c_n)+\limsup_n b_n \le \limsup_n (a_n+b_n)+\limsup_n (b_n+c_n)? $$
2
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1answer
28 views

Non-additive upper logarithmic density: $\ell^\star(X \cup Y) \neq \ell^\star(X)+\ell^\star(Y)$

Let $\ell^\star$ be the upper logarithmic density on the set of positive integers, namely $$ \forall X\subseteq \mathbf{N}^+, \,\, \ell^\star(X)=\limsup_n \frac{1}{\ln n}\sum_{x \in X\cap ...
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0answers
18 views

$\liminf_n \min(a_n,b_n)=\min( \liminf_n a_n, \liminf_n b_n)$

Do you have a reference for the following intuitive result? Let $(a_n)_{n\ge 1}$ and $(b_n)_{n\ge 1}$ be two sequence of reals. Then $$\liminf_n \min(a_n,b_n)=\min( \liminf_n a_n, \liminf_n b_n).$$
2
votes
4answers
66 views

Superior limit of a certain sequence

Let $(x_n)$ be the sequence: $$\{1, 2, 1 + \frac12, 2 + \frac12, 1 + \frac13, 2 + \frac13, \ldots \}$$ I (think I) understand why $\lim \inf x_n = 1$, but I'm not sure why $\lim \sup x_n =2$. Any ...
3
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0answers
36 views

$\liminf_n a_n = \inf_n a_n$ if $a_n \ge a_m$ when $n\mid m$

I would like to ask a reference for the following very easy result: can someone help? Let $(a_n)_{n\ge 1}$ be a sequence of positive reals such that $a_m \le a_n$ whenever $n$ divides $m$. Then $$ ...
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2answers
60 views

$\limsup = \liminf$ of sequence of Sets

This problem was on my in-class final for a measure theory course I took in the fall, and now I am studying for my qualifying exam so I am trying to figure this one out: Suppose ...
2
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1answer
50 views

Lim sup/inf of average value

Consider $$f(t)= \frac{1}{t} \int_{0}^t \sin(e^s) ds.$$ What is $$\mathrm{lim \ inf}_{t \rightarrow \infty} f(t)$$ and $$\mathrm{lim \ sup}_{t \rightarrow \infty} f(t)?$$ Using $u$-substitution, ...
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1answer
41 views

Prove that limsup and liminf of an independent sequence are independent of finite number of terms

Let $X_1, X_2, ...$ be an independent sequence of random variables on $(\Omega, \mathscr{F}, \mathbb{P})$. What I'm trying to prove is: Prove that $X_1, X_2, ..., X_k$ is independent of $\liminf ...
1
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2answers
63 views

Are random variables independent of their tail sigma-algebra?

Let $X_1, X_2, ...$ be independent random variables. Define $$\mathscr{T}_n = \sigma(X_{n+1}, X_{n+2}, \ldots)$$ and $$\mathscr{T} = \bigcap_{n} \mathscr{T}_n,$$ the tail σ-algebra of $(X_1, X_2, ...
0
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1answer
71 views

Applying Gronwall's Inequality

I've been working on this problem and was able to get it down to this inequality: $$E[\log X_t] \leq ae^{-t} + b(1-e^{-t}) + e^{-t}\int_0^t e^sE[\log X_s] ds$$ I'd like to use Gronwall's Inequality ...
1
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1answer
79 views

The value of limsup and liminf of a sequence of a sets obtained by combining three sequences

What is the limit superior of the following sequence of sets? $\{X_n\}=\{\{1/2\},\{1/3\},\{1/4\},\{2/3\},\{1/3\},\{1/5\},\{3/4\},\{1/3\},\{1/6\}......\}(n\to∞)$ I.e., $X_1=\{1/2\}, X_2=\{1/3\}, ...
3
votes
1answer
43 views

“Counterexample” to this characterization of lim sup?

I came across an exercise (Exercise 10, Ch. 1, Marsden's elementary classical analysis, 2nd ed.) that gives a characterization of lim sup I had never seen, which can be rephrased as follows: Let ...
2
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1answer
33 views

Are these implications true for a nonnegative stochastic process $X_t$?

Suppose I have a nonnegative stochastic process $X_t$. Furthermore, suppose the following is true: $$\limsup_t \frac{1}{t}E\left[ \log X(t)\right] \leq a < 0$$ for some constant $a \in ...
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0answers
17 views

Physical meaning about $\limsup$ of a Stochastic Process

I'm trying to show that a positive $n$-dimensional stochastic process $X_t = (x_1(t), \cdots, x_n(t))$ is nice in that it's well-behaved and controlled (in the sense that the process doesn't grow too ...
2
votes
1answer
52 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 ...
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0answers
20 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
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0answers
18 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
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0answers
54 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
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1answer
54 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
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3answers
36 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
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0answers
15 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( ...
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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
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2answers
38 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
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1answer
30 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 ...
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1answer
26 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
47 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.
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0answers
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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
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2answers
33 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
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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
49 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
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0answers
35 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
161 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
32 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
36 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
37 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
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3answers
30 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
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2answers
49 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 ...