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Questions tagged [limsup-and-liminf]

For questions concerning the definition and properties of limit superior and limit inferior of sequences of sets or real numbers.

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Let $(x_n)$ be a bounded sequence and $u=\limsup x_n$. Let E be set of limits of convergent subsequences of $(x_n)$. How do I prove $u \in E$?

I've been trying to attempt this problem for a long time now. At fist I tried to show that the sequence $(u_n)$, where $u_n = \sup_{i \geq n} x_i$, is a subsequence of $(x_n)$. But this is not true ...
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65 views

Edit: If $\liminf\limits_{n\to\infty}{a_n}>0$ then there exists $a\in \mathbb{R}$ such that $a_n\geq a>0$ for all $n\in \mathbb{N}$

Suppose that the $\beta=\liminf\limits_{n\to\infty}{a_n}>0$ where $\{a_n\}\subset(0,\infty)$ then there exists $a\in \mathbb{R}$ such that $a_n\geq a>0$ for all $n\in \mathbb{N}$. I've always ...
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33 views

$\liminf_\limits{n\to\infty}1_{A_n}(x)=1$ $\implies$ $\lim_\limits{n\to\infty}1_{A_n}(x)=1$?

Source: Partial proof from textbook: I've omitted the case where $x\in A^c$ as it's not relevant. I've also highlighted the part I'm having trouble with in blue. Here is my attempt at explaining ...
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limsup liminf duality

I'm reading this PDF authored by Philip Pennance. In (23) we have: given $(a_n)$, $a_n>0$, then $$\tag{*} \liminf \frac{a_{k+1}}{a_k} \le \liminf \sqrt[k]{a_k} \le \limsup \sqrt[k]{a_k} \...
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1answer
17 views

Example of sequence with interesting rate of growth

I am looking for an example of a sequence $\{a_n\}_{n\in \mathbb{N}}$, with $a_n \geq 0$, such that for $k<2$ $$ \limsup_{n\to \infty} \frac{a_n}{n^k} >0 ,$$ and that $$ \limsup_{n\to \infty} \...
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39 views

If a sequence $(u_n)$ be such that its every subsequence has a subsequence that converges to $0$, then $\lim u_n= 0$

Suppose that $(u_n)$ is unbounded above. Then, we pick any arbitrary monotone increasing subsequence $(v_n)$ of $(u_n)$. But by hypothesis, we can find a subsequence of $(v_n)$ that converges to $0$. ...
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62 views

Prove the sequence $\{x_n\}$ diverges to $+\infty$ if and only if..

Prove that the sequence $\{x_n\}$ diverges to $+\infty$ if and only if $$\limsup_{n\rightarrow\infty} x_n=\liminf_{n\rightarrow\infty} x_n=+\infty$$ My attempt: $(\rightarrow)$ By hyphotesis we know $...
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38 views

lim inf, lim sup, limit points

Let $t$ and $s$ be real numbers with $t<s$. Suppose that as $n \to \infty$, $a_{2n} \to t$ and $a_{2n-1}\to s$. True or False: It must be that $\liminf a_{n} = t$ and $\limsup a_{n} = s$ and $(...
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28 views

nth root of Bernoulli numbers

I'm trying to prove that the supremum limit is equal to infinity: $\limsup_{n->\infty}\sqrt[n]{|B_n|}=\infty$ Where $B_n$ is defined via the series expansion: $f(z)=\frac{z}{e^{z}-1}=\sum_{n=0}^{\...
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179 views

Correctly reading this $\lim \inf ()$ and $\lim \sup()$ expression?

I've never thought that I would have difficulties to read such a simple formula, which goes as follows1: A well-known unsolved problem in number theory concerns the distriubtion of $(3/2)^n\pmod1$. ...
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1answer
16 views

Regarding property of limit supremum of a bounded sequence $(x_{n})$

Let $X = (x_{n})$ be a bounded sequence. Limit superior of $X$ is $x^* = limsup(x_{n})$ then $\forall \epsilon > 0$ there exists atmost finite number of $n \in \Bbb{N}$ such that $x^* + \epsilon &...
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36 views

When does this sum Convergence?

Sequence $(a_n)_{n\in \mathbb{N}}$ such that $0 \leq a_n$ and $\sum_n a_n = 1$ . We also know that $a_n $ is non-zero for infinitely many n . For each $j \in \mathbb{N}$ we have non-negative sequence $...
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1answer
30 views

make an exponential curve tend to a specific value

I am trying to make the following curve to exhibit an exponential decline from a given value to zero. So far, I only managed to make it go from infinite to zero.. Thus I still need to make the curve ...
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112 views

Prove that $\limsup_{x\to\infty}\left(\cos x + \sin\left(\sqrt2 x\right)\right) = 2$

Prove that $$ \limsup_{x\to\infty}\left(\cos x + \sin\left(\sqrt2 x\right)\right) = 2 $$ Pretty much always when I ask a question here I do provide some trials of mine to give some background. ...
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1answer
24 views

Show that $\lim \inf x_n$ is an adherence value

Let $(x_n)$ be a bounded sequence. Show that $\lim \inf x_n$ is an adherence value of $x_n$. My proof: Let's define $a:=\lim \inf x_n$. Let $\epsilon > 0$ and $N \geq 1$. We'll show that $|x_{kn} ...
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1answer
23 views

Asymptotic behaviour of Integrals

Let $f,g$ be functions in $C_b\left(\left[\varepsilon,\frac{1}{2}\right]\right)$, $\forall \; \varepsilon > 0$, or equivalently $f,g \in C\left(0,\frac{1}{2}\right]$. We have $f(p) \overset{p \to ...
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38 views

Is there a way to state $\limsup_{n\to\infty}{x_n}=x$ using “\forall”

Denote $\{x_n\}$ as a sequence of real numbers, and there exist a $x\in\mathbb{R}$ such that $\limsup_{n\to\infty}{x_n}=x$. We can write "$\lim_{n\to\infty}x_n=x$" as "$\forall \epsilon>0,\exists N\...
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1answer
32 views

Slightly alternative proof to the converse part of Cauchy's General Principle

I want to prove that: If $\forall \epsilon >0$, $\exists k \in \mathbb{N}$, such that $| u_{n+p}-u_n| <\epsilon $, whenever $n\geq k$, $p\in \mathbb{N}$, then $\{u_n\}$ is convergent. Proof: [...
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36 views

A homework question about partial limits

I could really use some help figuring out this question. The question: ${a_n}$ is a series so that $\lim_{n\to\infty} (a_{n+1} - a_n) = 0$. Prove that its group of partial limits is the closed ...
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1answer
20 views

How to change the order of the limit and the expectation?

Does anyone know how to prove $\lim E[X(n)]=E[\lim X(n)]$??? Here I need to prove $\lim E[X(n)]\le E[\lim X(n)]$ and $\lim E[X(n)]\ge E[\lim X(n)]$. Based on "Fatou Lemma", I can get that $E[\...
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1answer
50 views

Limit superior and inferior when one part diverges

How can I find the limit superior and inferior of given sequence: $x_n = (1 + \frac{1}{2n})\cos{\frac{n\pi }{3}}$ as $ n \in \mathbb N $ I did the following: since $\lim_{n\to\infty}(1 + \frac{1}{2n})...
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1answer
36 views

Suppose $x_{n}>0$ and tends to zero.We know its arithmetic mean tends to zero. I wonder which will tend to zero faster.

In detail, $x_{n}>0$ and tends to zero, $S_{n}=\sum_{k=1}^{n}x_{k}$ (so $\lim_{n\to\infty}\frac{Sn}{n}=0$). Let $T_{n}=\frac{nx_{n}}{S_{n}}.$ I think the "average" $S_n/n$ will tend to zero more ...
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17 views

Root test of $\frac{1}{2n}$

Let $a_n=\frac{1}{2n}$ Then $|a_n|^\frac{1}{n}=\frac{1}{2n}^\frac{1}{n}$ And $\frac{1}{2n}^\frac{1}{n}<1 $ for every $n\geqslant1$. So $L=\limsup_{n\rightarrow\infty} |a_n|^\frac{1}{n}<1$ By ...
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1answer
27 views

If $c_{n}=\frac{n\cos(n\pi)}{n+1}$, find $\limsup c_{n}$ and $\liminf c_{n}$.

If $c_{n}=\frac{n\cos(n\pi)}{n+1}$, find $\limsup c_{n}$ and $\liminf c_{n}$. Here is what I did: $\left \{c_{n} \right \}=\left \{-\frac{1}{2},\frac{2}{3},-\frac{3}{4},\frac{4}{5},-\frac{5}{6},... ...
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1answer
15 views

Sequence definition of Limit Sup

Cannot understand the sequence definition of the limit sup of a sequence ($a$n) How can we say that $a$n < Lim sup + € except finitely many terms ( Why cant we say that $a$n< lim sup +€ for ...
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2answers
19 views

limit superior and inferior 2

Let $x_n$ be a sequence. If $M =\limsup x_n$ then there is some subsequence $x_k$ with $M = \lim x_k$. Then $$ \left|x_k - M \right|< \epsilon \iff M - \epsilon < x_k < M+\epsilon \quad \...
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1answer
34 views

Limit sup limit inf

I am getting confused about the definition of $\limsup$. It's been given in the books that if $M$ is the $\limsup x_n$ then $x_n < M+\varepsilon$ except finitely many $n$ i.e. for all $n > N$ ...
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2answers
55 views

limit inferior and infimum (real analysis)

I want to prove $$\liminf_{n \to \infty} \left( \inf_{x \in X} f_n(x) \right) \leq \inf_{x \in X} \left( \limsup_{n \to \infty} f_n(x)\right)$$ when $X \subset \mathbb{R}$ and $f_n\colon X \...
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67 views

Given $\lim_{n \rightarrow \infty} a_n a_{n+1} = L$, how to show: $\lim_{n \rightarrow \infty} a_n a_{n+3} = L$

let $\{ a_n \}$ be a sequence where for each $n \in \mathbb N$ $ a_n \neq 0 $ and where $\lim_{n \rightarrow \infty} a_n a_{n+1} = L$ with $L \neq 0$ I want to prove that $\lim_{n \rightarrow \...
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1answer
67 views

Example $f_n = \mathbf{1}_{[n,\infty)}$ in Fatou's Lemma

In the class, my professor gave the following example: Let $f_n = \mathbf{1}_{[n,\infty)}$, then we have $$\int_X \lim\inf f_n = 0, \text{ since } \lim\inf f_n = 0,$$ and $$\lim\inf \int_X f_n = ...
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0answers
32 views

Proving property of $\liminf_{x \to x_0} f(x)$

Let $f: X \to \mathbb R$. Define $$\liminf_{x \to x_0} f(x) = \lim_{r \to 0^+} \inf_{x \in B_r(x_0)} f(x)$$ Show if $x_n \to x_0$ then $$\liminf_{n \to \infty} f(x_n) \ge \liminf_{x \to x_0} f(x)$$ ...
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3answers
156 views

How can we compute $\liminf_{n→∞}(x_n+y_nz_n)$ for monotone $(x_n)_{n\in\mathbb N}$ and $(y_n)_{n\in\mathbb N}$ knowing that $\liminf_{n→∞}z_n=-1$?

Let $(x_n)_{n\in\mathbb N},(y_n)_{n\in\mathbb N},(z_n)_{n\in\mathbb N}\subseteq\mathbb R$ with $$\liminf_{n\to\infty}z_n=-1\tag1.$$ I want to show that if $(x_n)_{n\in\mathbb N}$ is ...
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Question about lim inf or lim sup

When taking the course of real analysis, the professor wrote the following: Let $f_n \geq 0$ for all $n$. Let $$a = \lim \inf \int_X f_n d\mu + \epsilon$$ I think here $\epsilon$ is any positive ...
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55 views

Show that the limit inferior of the sum of third moments of mutually independent Gaussians tends to negative infinity

Let $d\in\mathbb N$, $\sigma:=\ell d^{-\alpha}$ for some $\ell>0$ and $\alpha>0$, $Y$ be a Gaussian $\mathbb R^d$-valued random variable with mean $0$ and covariance matrix $\sigma^2I_d$$^1$, $Z$...
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2answers
49 views

Suppose $(a_n)^{n\to\infty}_{n=1}$ is a bounded sequence of real numbers. Prove that: $\liminf a_n \leq\limsup a_n$

Suppose $(a_n)^{n\to\infty}_{n=1}$ is a bounded sequence of real numbers. Prove that: $$\liminf a_n \leq\limsup a_n$$ This makes sense as $\inf a_n$ is the lowest bound of $a_n$ and $\sup a_n$ is ...
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1answer
28 views

Logical proof for independent sequence of sets

I need to show that: If $(E_n), n\geq 1$ and $(F_n), n\geq 1$ are non-decreasing sequences which converge respectively to $E$ and $F$, and $E_n$ and $F_n$ are independent, then $E$ and $F$ are ...
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1answer
52 views

Problem to understand Borel-Cantelli Lemma

I needed to prove the Borel-Cantelli lemma, which states that If $$ \sum_{n=1}^{\infty} P(A_n) < \infty \implies P(\limsup A_n) = > 0,$$ $$\sum_{n=1}^{\infty} P(A_n) = \infty \implies P(...
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1answer
43 views

Can we conclude $\liminf_{n\to\infty}\operatorname E\left[1\wedge e^{X_n}\right]=0$ from $\liminf_{n\to\infty}X_n=-\infty$?

Let $(X_n)_{n\in\mathbb N}$ be a real-valued process on a probability space $(\Omega,\mathcal A,\operatorname P)$ with $$\liminf_{n\to\infty}X_n=-\infty\;\;\;\text{almost surely}.\tag1$$ Are we able ...
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2answers
61 views

How to evaluate $\lim_{N\to\infty}\sup_{x\in(1,\infty)}\big|\sum_{n=N+1}^{\infty}\frac{1}{1+x^n}\big|$?

How to evaluate the following limit $$\lim_{N\to\infty}\sup_{x\in(1,\infty)}\big|\sum_{n=N+1}^{\infty}\frac{1}{1+x^n}\big|\ ? $$ My hunch is that this limit should be zero, but I am unable to prove ...
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57 views

$\liminf$ and $\limsup$ are in $\sigma$-algebra?

Let $F$ be a $\sigma$-algebra of sets of $\Omega$. I have to show that if $(A_n)$ for $ n \geq 1$ is a sequence of events such that $A_n \in F$, then $\liminf A_n$ and $\limsup A_n$ are both elements ...
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2answers
50 views

Finding the lim inf and lim sup of a sequence

I'm working on the following problem: Find $\lim \sup_n a_n$ and $\lim\inf_n a_n$ of the sequence given by $a_n = 1 + (-1)^n\frac{2n+3}n.$ I've done the following work so far: Let $\{i_n\}_n$ be ...
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2answers
49 views

Set sequences $\liminf$ and $\limsup$ - correct?

I'm studying set theory for probability and statistics, and it's important, in order to work with a $\sigma$-algebra, to discuss the concept of $\liminf$ and $\limsup$ for sequences of sets. But in ...
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3answers
72 views

What does this notation mean: $\limsup_{\epsilon \to 0} \dots$

In Cohn's measure theory, second edition, p166, there is written: Let $\mu$ be a finite Borel measure on $\mathbb{R}^d$. Then the upper derivate $(\overline{D}\mu)(x)$ of $\mu$ at $x$ is defined by ...
3
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0answers
62 views

Very challenging measure theory problem with $L^p$ norms and lower bound for probability [closed]

In a probability space X, we are given $f_n$ and $a>0$ so that $1\le\|f_n\|_2\le {1 \over a} \|f_n\|_1$ for every $n\ge 1$. (The $L^2$ and $L^1$ norms respectively) Prove that $\:P(|f_n(x)|\ge a/...
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1answer
65 views

$\lim$ vs $\liminf$ and $\limsup$ in the proof convergence in probability implies convergence in distribution

I am studying the various types of convergence for random variables, in particular how convergence in probability implies convergence in distribution. Let $(X_n)_n$ be a sequence of random variables ...
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3answers
163 views

Is it possible to find $\limsup\limits_{n\to\infty} \frac{2×3^n-3×2^n}{2^{\alpha(n)}-3^n}$?

I need help to solve this problem: $$\lim_{n\to\infty}\frac{2\cdot 3^n-3\cdot 2^n}{2^{\alpha(n)}-3^n}$$ or $$\limsup_{n\to\infty}\frac{2\cdot 3^n-3\cdot 2^n } {2^{\alpha(n)}-3^n}$$ where, $\...
1
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2answers
47 views

If $a_n=n\left(1-\frac{1}{n}\right)^{n[\log n]}$, prove $1\leqslant \liminf a_n$ and $\limsup a_n\leqslant e.$

Prove that: $$\limsup n\left(1-\frac{1}{n}\right)^{n[\log n]} \leqslant e$$ and $$\liminf n\left(1-\frac{1}{n}\right)^{n[\log n]} \geqslant 1.$$ Attempt. Since for $n>2$: $$1<[\log n]&...
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1answer
40 views

Greatest lower bound of the set : $\{(e^n + 2^n)^\frac1n\ | \; n\in \mathbb{N}\}$

Find the greatest lower bound of the set : $\{(e^n + 2^n)^\frac1n \mid n\in \mathbb{N}\}$ I will find the limit of $a_n := (e^n + 2^n)^\frac1n $ which if exists, must equal the greatest lower ...
0
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1answer
101 views

Convergence of $a_n=\sum\limits_{m=2}^{n} \frac{(-1)^mm}{(\ln(m))^m}$ and $b_n = \sum\limits_{m=2}^{n} \frac{1}{(\ln(m))^m}$

Let $a_1=b_1=0$ and for each $n\geq 2$, let $a_n$ and $b_n$ be real numbers given by $a_n=\sum\limits_{m=2}^{n} \frac{(-1)^mm}{(\ln(m))^m}$ and $b_n = \sum\limits_{m=2}^{n} \frac{1}{(\ln(m))^m}$,...
2
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1answer
29 views

Let $(a_n),(b_n) $ be sequences of positive real numbers such that $na_n<b_n<n^2a_n$

Let $(a_n),(b_n) $ be sequences of positive real numbers such that $na_n<b_n<n^2a_n$ for all $n\geq2$. If the radius of convergence of $\sum a_n x^n$ is $4$, then $\sum b_n x^n$ A)converges ...