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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Limit Superior and Inferior of a Sequence [closed]

Let Pn be a bounded sequence of real numbers and let p∈R be such that every convergent subsequence of Pn converges to p. Prove that the sequence Pn converges to p.
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31 views

Any bounded sequence in $\mathbb R$ has a smallest cluster point $\underline x$ and a greatest cluster point $\overline x$

Theorem: Any bounded sequence in $\mathbb R$ has a smallest cluster point $\underline x$ and a greatest cluster point $\overline x$ and they satisfy $\liminf x_{n} = \underline x$ and $\limsup x_{n}=\...
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26 views

Asymptotic equivalence and lim inf, lim sup

Given two positive real sequences $(U_n)$ and $(V_n)$ with $(V_n)>0$ and two fixed real numbers $a$ and $b$, is it true that if $\underset{n \rightarrow +\infty}{\lim \sup} \text{ }\frac{U_n}{V_n} ...
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41 views

Convergence of a sequence of sets $A_n:=\{1+ \frac{m^2}{n^2}: m \in \mathbb{N} \}$

Let $n \in \mathbb{N}$ and let $A_n$ be the set given by $$A_n:=\left\lbrace 1+ \frac{m^2}{n^2}: m \in \mathbb{N} \right\rbrace$$ Can you help me to determine if $\lim_{n \to \infty} A_n $ exists? ...
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1answer
17 views

What is the definition of a limit superior for a set valued mapping?

I'm considering a set valued mapping $X(t): \mathbb{R} \to \mathcal{P}(\mathbb{R}^n)$, where $\mathcal{P}$ denotes the power set. Given a paramater $t \downarrow 0$, I thought I could define the $\...
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2answers
117 views

Is there a sequence $x_n\to+\infty$ such that $\liminf x_{2n}/x_n = 0$?

I have a sequence of real numbers $(x_n)$ that diverges to $+\infty$. Can I conclude somehow that $$\liminf \frac{x_{2n}}{x_n}>0,$$ or are there counterexamples?
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1answer
33 views

$X_n \xrightarrow{n \to \infty} X$ almost surely implies $\mathbb{E}[|X|] \le \liminf_{n \to \infty} \mathbb{E}[| X_n |]$

Not a homework question but an exercise from an past exam. Let $X, X_1, X_2, \ldots$ be real-valued random variables on a measure space $(\Omega, \mathcal{F}, \mathbb{P})$. Show that $X_n \...
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136 views

Prove that $A_n \cap B_n \rightarrow A \cap B$

If $A_n \rightarrow A$ and $B_n \rightarrow B$ are sequences of sets then is it true that $A_n \cap B_n \rightarrow A \cap B$? How to prove or provide a counterexample? I had thought the following ...
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67 views

$\limsup$ and $\liminf$ of family of sequences with given properties

This is from a practice exam for my quals. Let $(x_n)_{n=0}^\infty$ be an arbitrary sequence in $\mathbb R$ satisfying: $x_n \ge 0$ $x_n + 2x_{n+1} \le 6$ for all $n \ge 0$ Part $b)$ is ...
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61 views

A seemingly true claim?

Let $(a_n)$ be a sequence of positive real numbers. If $(a_n)$ is not bounded then $\lim\sup\Big(\frac{a_n}{1+a_n}\Big)=1$. I was tempted to make the above claim when I tried proving the result below:...
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1answer
19 views

If $f(s) = g(\frac{1}{s})$, then $\limsup\limits_{s \rightarrow \infty} f(s) = \limsup\limits_{m \rightarrow 0+} g(m)$. [closed]

Assume that $f$ is a function from $(0, \infty)$ to $\mathbb{R}$. Then we can define a function $g : (0, \infty) \rightarrow \mathbb{R}$ satisfying $f(s) = g(\frac{1}{s})$. In this case, the ...
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31 views

Equivalence of a lim sup of functions with a lim inf of sets.

$\{\omega\in\Omega : \limsup_{n\to\infty}\lvert Y_n(\omega)-Y(\omega)\rvert=0\}$ =$\liminf_{n\to\infty}\{{\omega\in\Omega:\lvert Y_n(\omega)-Y(\omega)\rvert\lt \epsilon\}}\forall\epsilon>0$. How ...
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37 views

Probability of Brownian Motion

I've studied a proof from the book An Introduction to Stochastic Differential Equations concerning the nowhere differentiability of the Brownian motion and I'm stuck at the following proof: In the ...
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1answer
77 views

Given $\limsup \frac{a_n}{b_n} < \infty$. Prove there is a constant M such that $a_n \leq Mb_n$ [duplicate]

Given sequences $\{a_n\}$ , $\{b_n\}$ of positive real numbers and $\textrm{ limsup } \frac{a_n}{b_n} < \infty$. Prove there is a constant M such that $a_n \leq Mb_n$ Defn: Let $(a_{n})_{n=...
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32 views

ε environment of number 1

I am stuck with a problem it asks me to find the $\epsilon$ environment of the number $X=1$ and find for which $\epsilon$. I got a few intervals to solve for, here are the intervals: $(0,2) , (\frac{2}...
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39 views

What is $\lim \sup_{n \to \infty} \sum_{k=1}^{n} (-1)^k \left(\frac{1}{2} -\frac{1}{k} \right)$?

Let $a_{n}=(-1)^n \left( \dfrac{1}{2} -\dfrac{1}{n}\right)$ and let $b_n=\sum\limits_{k=1}^{n} a_{k}, \forall \ n \in \mathbb{N}$. Then what is $\lim \sup b_{n}$ and $\lim \inf b_{n}$? $\begin{...
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16 views

Convolution of two iid random variables is lower semicontinuous

I have a question regarding two iid random variables $X_1$ and $X_2$, each of which being a copy of the non-negative rv $X$ with $\lambda^1$-density function $f:\mathbb{R} \rightarrow [ \, 0, +\infty \...
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1answer
50 views

Proof that $a_n \leq b_n$ implies that $\limsup a_n \leq \limsup b_n$.

I am trying to prove that for sequences $(a_n)$ and $(b_n)$, that if $a_n \leq b_n$ for all $n \geq m$, then $\limsup\limits_{n \to \infty} a_n \leq \limsup\limits_{n \to \infty} b_n$. This is part ...
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1answer
73 views

Tao Lemma 6.4.13.

I am trying to prove this following lemma in Tao's analysis text. Suppose that $(a_n)_{n=m}^{\infty}$ and $(b_n)_{n=m}^{\infty}$ are two sequences of real numbers such that $a_n \leq b_n$ for all $...
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4answers
73 views

Radius of convergence of power series $\sum _{n=1} ^{\infty} \frac{n!}{n^n} x^{2n}$

Radius of convergence of power series $$\sum _{n=1} ^{\infty} \frac{n!}{n^n} x^{2n}$$ I have calculated $\overline\lim a_n^{\frac{1}{n}}= \infty $. So radius of convergence should be zero. But ...
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131 views

Limit of a power series in $\beta$ multiplied by $(1 - \beta)$

Suppose that you are given a bounded sequence of real numbers $|w_k| \le W$. What should be the limit $\lim_{\beta \rightarrow 1^-}\ (1 - \beta) \sum_{k = 0}^\infty \beta^k w_k$? To see that the ...
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34 views

If $\limsup\limits_{n\rightarrow \infty} a_n=a< \infty$, then $\forall\epsilon>0$ $\exists N\in \mathbb{N}:a_n\leq a+\epsilon$

If $\limsup\limits_{n\rightarrow \infty} a_n=a< \infty$, then there exists for all $\varepsilon >0$ a $N\in \mathbb{N}$ such that $a_n\leq a+\varepsilon$ for all $n\in \mathbb{N}$, $n\geq N$ ...
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2answers
66 views

If there's only one limit point, what's $\limsup_{n\to \infty} a_n$ and what's $\liminf_{n\to \infty} a_n$?

$a_n:=(-1)^n\cdot 1/n-2n^2(1+(-1)^n)+n/(n+1)$ with $n\in \mathbb{N}$. I came to the conclusion that there's just a single limit point. If $H$ is the set of Limit points, it's $H=\{1\}$. How do I ...
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50 views

Prove that $\limsup_{n\to\infty}(a_n)\le \sup(a_n)$ [duplicate]

Let $a_n$ be a bounded sequence. Prove that $$\limsup_{n\to\infty}(a_n)\le \sup(a_n)$$ So I think that the best way to prove it is assume that $\limsup_{n\to\infty}(a_n)> \sup(a_n)$ and then find ...
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33 views

Simple Liminf Question for a Bounded Sequence

Consider a sequence $\{a_n\}$ satisfying $ a_{\min} \leq a_n \leq a_{\max}$ with $-1 <a_{\min}<0<a_{\max}$. Now, consider a new sequence $$ s_n := - (1+a_{n-1})a_n $$ I would like to find ...
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1answer
38 views

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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1answer
66 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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1answer
38 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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19 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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2answers
44 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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64 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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2answers
39 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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2answers
36 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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187 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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0answers
37 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
32 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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1answer
117 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
25 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
24 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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2answers
40 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
33 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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2answers
37 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
21 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[\...
4
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1answer
51 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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2answers
18 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
28 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
16 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
23 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 \...