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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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How to understand that both the lim inf and lim sup of the sets of real numbers is either open or closed?

I am reading the Real Analyais (4th edition) by Royden, H. L., & Fitzpatrick, P. In Section 1.4, it is said that both the lim inf and lim sup of a countable collection of sets of real numbers, ...
Lau's user avatar
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2 answers
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Prove convergence of $\limsup_{n\to\infty}$

I am new to Real Analysis, and I have found this problem hard to formalize. Problem Let $(p_n)_{n\in\mathbb{N}}$ and $(q_n)_{n\in\mathbb{N}}$ sequences such that $(p_n)\to u$ and $(q_n)\to v$. ...
afm95's user avatar
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3 answers
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Why does $\lim(\inf (a_n +b_n))=\lim( \inf(a_n) )+ \lim (\inf(b_n))$ fail?

I was working on the proof that for bounded sequences $a_n$ and $b_n$, $\liminf_{n\to\infty}(a_n)+\liminf_{n\to\infty} (b_n) \leq \liminf_{n\to\infty} (a_n+b_n)$. I got to a point where I concluded ...
user1345605's user avatar
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Confusion about $\lim\sup$ and its definition as the greatest limit point

I posted a question a few days ago and the most voted answer uses $\lim \sup$, a concept I was not familiar with. I decided to jump ahead and read about $\lim\sup$ to understand the answer, but one ...
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Equivalent Definitions of Limsup [proof verification]

I've been working on this proof for a few days. While there've been several posts on this, none is not up to my standard for rigor. Below is my attempt, which seems clear to me except for the case ...
n1lp0tence's user avatar
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Prove that $\liminf_i a^{-1}_{i-j}a_i = \liminf_i a_i^{j/i}$ for any sequence $a_i$?

Title pretty much says it all. I saw this claim in some of my lecture notes which says this is a "general fact about sequences". In full: let $(a_i)_{i \in \Bbb{N}} \subset \Bbb{R}_{> 0}$ ...
soggycornflakes's user avatar
2 votes
2 answers
78 views

intuition on lim sup and lim inf in probability spaces

So I thought I understand what the lim sup and lim inf are, especially thanks to this post, but now I am presented with a problem from basic probability theory saying: Let $(A_n)$ be a sequence of ...
arridadiyaat's user avatar
1 vote
0 answers
49 views

About Uniqueness Proofs

My understanding of existence and uniqueness results is that the former condition asserts the cardinality of the "solution set" is positive, whereas the latter implies it is bounded above by ...
n1lp0tence's user avatar
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1 answer
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$\lim \sup$/ $\lim \inf$ inequalities fail when $\mu(X)=\infty$

I think this question was already answered in multiple places, e.g. here, but there is still one thing I cannot grasp: Consider the measure $\mu$ and a sequence of sets $A_n$. Given the condition $\mu\...
arridadiyaat's user avatar
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$\inf_{n\in \mathbb{N}^*} \left\{\frac{3^n}{2^n} \right\} > 0$?

The fractional part is defined of a positive real number $x$ is defined by $\{x\}:=x-\lfloor x\rfloor$. Is it true that $$\inf_{n\in \mathbb{N}^*} \left\{\frac{3^n}{2^n} \right\} > 0 \ ?$$ If so, ...
Nathan Portland's user avatar
2 votes
1 answer
37 views

Characterisation of $\limsup$ of a function

Consider $\Omega\subset\mathbb{C}$ open, $z^*\in\Omega$ and $f:\Omega\setminus\lbrace z^*\rbrace\rightarrow\mathbb{C}$. Then $\limsup_{z\rightarrow z^*}|f(z)|:=\inf_{r>0}\sup_{z\in B_r (z^*)\cap\...
ramind's user avatar
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1 answer
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Construct a sequence $(a_n)_{n=1}^{\infty}$ which has exactly three limit points, at $- \infty, 0,$ and $+ \infty$

Definition. Let us say that a sequence $(a_n)_{n=M}^{\infty}$ of real numbers has $+\infty$ as a limit point iff it has no finite upper bound, and that it has $-\infty$ as a limit point iff it has no ...
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Existence of a subsequence that has positive limit, given the original sequence is bounded and has a positive limit superior.

Suppose I have a sequence $f: \mathbb N \rightarrow \mathbb R$ such that $0 \le f(n) \le 1$ for all $n$. By the Bolzano Weierstrass theorem, that there must be convergent subsequences of $f$. Suppose ...
PD_Sathya's user avatar
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Definition of $\limsup$ from the right or from the left

Let $X$ be a metric space and $E \subset X$. If $f: E \rightarrow \mathbb{R}$, we can define for $a \in E$, the superior limit \begin{align} \limsup_{x \to a} f(x) = \lim_{r \to 0} \left( \sup\{ f(x) ...
Hilbert's Minion's user avatar
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1 answer
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Proving that if $L^+$ is finite, then it is a limit point of $(a_n)_{n=m}^{\infty}$ Analysis I, Terence Tao

The below result comes from Analysis I, 3rd edition, Terence Tao. I attempted to prove part (e) below (I added parts (b) and (c) as I use them in the proof): Propsition 6.4.12. Let $(a_n)_{n=m}^{\...
Paul Ash's user avatar
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3 votes
1 answer
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Explain step in proof of $\lim_{n \to \infty} a_n = b \iff \limsup_{n \to \infty} a_n = \liminf_{n \to \infty} a_n = b$.

I'm trying to understand a step in the forward direction of the proof of the theorem $\lim_{n \to \infty} a_n = b \iff \limsup_{n \to \infty} a_n = \liminf_{n \to \infty} a_n = b$. First, to clarify, ...
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How to explain the relation between upper/lower limits of sequence of real numbers and sets?

Please be patient with my English as my first language is not English. I'm currently learning set theory and I find myself puzzled about the relationship between the definitions of upper/lower limit ...
chenyq's user avatar
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4 votes
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Finding $\liminf n (\sin n)^2$

I'm solving an exercise from The elements of Real Analysis by Robert G. Bartle, which asks to find the $\limsup$ and $\liminf$ of the sequence given by $a_n = n (\sin n)^2$. I have already calculated $...
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Questions About Four Definitions of The Upper and Lower Limits of A Sequence

Related questions have been posted here and here. Background I have seen the following four definitions of the upper and lower limits of a sequence from textbooks and MSE posts: Definition 1$\quad$ [...
Beerus's user avatar
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Expressing the event $ \{ \limsup_{n \rightarrow \infty} X_n = l \}$

Let $(\Omega, \mathcal{H}, \mathbb{P})$ be a probability space. Let $(X_n)_{n=1}^{\infty}$ be a sequence of real valued random variables s.t. $\forall n \geq 1 \quad X_n$ is $\mathcal{H}$-measurable. ...
Fran712's user avatar
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The limit points of the sequence $x_n=1+\sin\left( \frac{n\pi}{4} \right)$

Consider the sequence \begin{align} x_n= 1 + \sin\left(\frac{n\pi}{4} \right). \end{align} I need to find the limit points of $\lbrace x_n \rbrace$. Notice that \begin{align} x_{8n}&= 1 , \\ x_{8n+...
Hussein Eid's user avatar
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1 vote
2 answers
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Why is $\{\omega \in \Omega | lim_{n\to\infty}X_n(\omega)=1\}$ in the tail $\sigma$-Algebra?

On a probability space $(\Omega, \mathcal{A}, P)$ I have a series of random variables $ X_n, n \geq 1$ and $ \mathcal{A}_{\infty}:=\limsup_k \sigma(X_k)$ denotes the corresponding tail $\sigma$-field....
Lu1998's user avatar
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Asymptotic Behaviour of $e^{ \alpha \log{( \frac{a}{b})} \log(n)} \times \sum_{t=1}^{n/2} e^ {- \frac{2t}{3} (\log(t) - 3) }$

If $a>b>0$ and $\alpha = \frac{x+1}{2y}$ where $y>0$ and $x\geq 0$. Under which conditions the following summation is asymptotically ($n\to \infty$) upper-bounded by $ k n$ where $k$ is a ...
Jay's user avatar
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What is $\limsup f(x)$ as $x\rightarrow \infty$

Trying to solve Exercise $2.6$ from Real Analysis text by Shakarchi. Integrability of $f$ on $\mathbb{R}$ does not necessarily imply the convergence of $f(x)$ to $0$ as $x \to\infty$. There exists a ...
Mahammad Yusifov's user avatar
-1 votes
1 answer
33 views

Proving that $\liminf_{n\rightarrow\infty} \log X_n/\log n \le -1$ a.s. where $X_n$ are i.i.d. random variables, uniformly distributed on $[0,1]$

Let $X_n$, $n\in \mathbb{N}$ be independent random variables, uniformly distributed on $[0,1]$. Show that $$\liminf_{n\rightarrow\infty} \frac{\log X_n}{\log n} \le -1 \ \ \text{a.s.}$$ I added my ...
Ata Keskin's user avatar
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0 answers
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How to prove Exercise 8.2.6 from Analysis 1 Terence Tao [duplicate]

I have been stuck a while on the following exercise of Analysis 1 from Terence Tao: Exercise 8.2.6 Let $\sum^\infty_{n=0}a_n$ be a series which is conditionally convergent, but not absolutely ...
Smogogole's user avatar
1 vote
1 answer
55 views

Find $\mathop{\limsup}\limits_{n\to\infty} a_n$ and $\mathop{\liminf}\limits_{n\to\infty} a_n$ if $a_n = n(2+(-1)^n)$

Given the sequence $a_n = n(2+(-1)^n)$. Find $\mathop{\overline{\lim}}\limits_{n\to\infty} a_n$ and $\mathop{\underline{\lim}}\limits_{n\to\infty} a_n$. The following is how I approach this problem. $\...
dienhosp3's user avatar
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0 answers
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Lim Inf and Lim Sup of Double Sequences

Given a double sequence $(a_{n,m})_{n,m \in \mathbb{N}}$ with $0 \leq a_{n,m} \leq C$ for some constant $C > 0$, and knowing that $\liminf_{n \to \infty} a_{n,m} = 0$ for any fixed $m$, does it ...
Maff's user avatar
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0 votes
0 answers
37 views

Root Test Stronger than Ratio Test using $\lim \sup$ and $\lim \inf$

Let $\{a_n\}$ be a sequence of real positive numbers. We wish to prove that $$ \lim \inf \frac{a_{n+1}}{a_n} \leq \lim \inf a_n^{1/n} \leq \lim \sup a_n^{1/n} \leq \lim \sup \frac{a_{n+1}}{a_n}$$ The ...
monkey king's user avatar
1 vote
1 answer
93 views

Property of lower semicontinuous functions

I am trying to prove that f is sequentially lower semicontinuous at x if and only if $f(𝑥)=\sup_{r>0}\inf_{y \in B(x,r)}f(y)$. Following the proof of ($F$ is lower semicontinuous $\iff F(x)=\sup_{...
Sharon Puthuparambil's user avatar
1 vote
1 answer
38 views

Is the limsup perserved under monotonically increasing functions?

I've done a bit of digging but I haven't found any sources to confirm that this is true. Take $\limsup{\{x_n\}}=L$ and $f(x): D \to \mathbb{R}$ to be a continuous and monotonically increasing function ...
EzTheBoss 2's user avatar
3 votes
1 answer
50 views

Calculate the limit $\lim_{n \to \infty} \int_1^e x^m e^x (\log x)^n dx$ with limsup and liminf

I have a question about how to prove the limit using $\limsup$. On the other day, I was asked the following problem in the exam: Let $m$ and $n$ be positive integers. Calculate the following limit, ...
kuHamrry's user avatar
1 vote
1 answer
54 views

$\limsup_{n\rightarrow \infty} \frac{x_n}{y_n} = 1$ implies $x_n \geq (1-\epsilon)y_n$

Suppose $x_n$ and $y_n$ are a sequence of real numbers such that $$\limsup_{n\rightarrow \infty} \frac{x_n}{y_n} = 1 \quad \mathrm{and} \quad \liminf_{n\rightarrow \infty} \frac{x_n}{y_n} = -1.$$ ...
CBBAM's user avatar
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1 vote
0 answers
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Proof clarification

The book I'm reading showed a the promise without proving it, after the bolanzo Weirestrass theorem. The theorem is : Let $\left\{{a_n}\right\}^\infty _{n=1}$ , $\left\{{b_n}\right\}^\infty _{n=1}$ be ...
Someguyalive's user avatar
2 votes
2 answers
42 views

Exchanging limit and inferior limit

Let $b(k,M)$ be a real sequence such that for any $k$, $b(k,M)\in [-M,M]$. I know that $\lim_{M\to+\infty} \sup_{k\geq0} |b(k,M)-a(k)|=0 $, meaning that $b(k,M)$ converges to $a(k)$ when $M\to+\infty$...
Jaz's user avatar
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0 answers
87 views

lim sup, lim inf, when sequence converges

I found this question online (not homework) and wanted to check my answer. Let $$A_1 = \{r: r \textit{ rational, and } \lim \text{inf } x_n <r < \lim \text{sup } x_n \}$$ $$A_2 = \{r: r \textit{ ...
Cole's user avatar
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2 votes
1 answer
31 views

Existence of a limit from limsup and liminf

Consider a sequence of random variables $(p_n)$ such that: (1) $\Pr\bigg(\limsup_{n \to \infty} p_n\leq U\bigg)=1$ (2) $\Pr\bigg(\liminf_{n \to \infty} p_n\geq L\bigg)=1$ where $L,U$ are real numbers. ...
Star's user avatar
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1 vote
1 answer
137 views

Solving a pretty difficult convergence for series

Let $P_n(x)=x^n-nx+1$ be a sequence of polynomials, where $P_n\colon[1, +\infty) \to \mathbb{R}$ and $n \ge 2$ a) Show that for each $n$, $P_n(x)=0$ has exactly one solution, and for each $n$ let $...
Shthephathord23's user avatar
0 votes
1 answer
75 views

Some doubt on converges and divergences series [closed]

One of tool for determinate the behaviours of the series is to check if $$\limsup_{n\to\infty} \sqrt[n]{a_n}<1$$ or $$\liminf_{n\to\infty} \sqrt[n]{a_n}>1$$ But when the sequences $a_n$ ...
afraidguy 's user avatar
-2 votes
1 answer
70 views

Deducement of First and second Borel-Cantelli Lemma

Suppose that $\Omega$ is a set, $(\Omega, \mathscr{G})$ is a measure space, and $Z: \Omega \to \mathbb{R}$ is a given mapping. Then Z is $\mathscr{G}$ measurable iff $$Z =\displaystyle\sum_{i=1}^\...
Win_odd Dhamnekar's user avatar
1 vote
1 answer
90 views

Confusion on Baby Rudin Chapter 3 Exercise 14 (e)

The question posed is as follows: For $\{s_n\}$ a sequence of complex numbers, $\sigma_n = \frac{s_0 + s_1 + ... + s_n}{n+1}$, $a_n = s_n - s_{n-1}$, $| n a_n | \leq M < \infty$, $ \forall n \in \...
BBadman's user avatar
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0 votes
1 answer
138 views

Why $\limsup$ in the strong law of large numbers?

Let $(X_n)_{n \in \mathbb{N}}$ be an $L^1$ sequence of random variables that are independently and identically distributed with mean $\mu$. Let $$S_n = \sum_{i=1}^n (X_i - \mathbb{E}[X_i]).$$ The ...
CBBAM's user avatar
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0 votes
0 answers
31 views

A class of sequences of real numbers making unions for the unit open interval

We are dealing with the couples $(\beta,\{b_n\})$, were $\beta$ is a fixed real number and $b_n$ a sequence of real numbers, such that $$ \bigcup_{n=1}^\infty [b_n,\beta b_n)=(0,1) $$ A necessary ...
M.H.Hooshmand's user avatar
2 votes
0 answers
48 views

Measurability of $\limsup$ over an uncountable index set

In the stochastic processes course I am taking, the following result was stated without proof. Claim. $\quad$ Suppose that $B_t$ is a standard brownian motion with respect to the filtration $(\mathcal{...
Harry Partridge's user avatar
1 vote
0 answers
29 views

Proving inf sup (lim x->y f(xn)) = sup(lim(f(x_n->y))) [closed]

Can someone provide insight into proving this statement: inf sup (lim x->y f(x_n)) = sup(lim(f(x_n-)))
ABT 15's user avatar
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0 votes
1 answer
34 views

Strict inequality

Consider the measure space ($\Omega, \mathcal{A}$), $A \in \mathcal{A}$. Let $f_n: A \rightarrow [-\infty; \infty]$ be a sequence of measurable functions. My script claims $$\{x \in A: \inf_n \sup_{k \...
PesFAs2's user avatar
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0 answers
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Constructing the reals using limit of a sequence of sets containing rationals

This is a follow-up to the question: [1] Paradox: Creating an uncountable set of natural numbers. There is also a relation to this q: [2] Limits of sequences of sets. Are the limit points always ...
Anders H's user avatar
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1 answer
70 views

limsup/liminf of sets - probability

Trying to get a better understanding of limsup/liminf in terms of probability. An intuitive explanation of the limsup of a sequence of sets $(A_n)_{n\geq 1}$, in the context of probability, is that $\...
b.b.89's user avatar
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1 vote
1 answer
86 views

Limit superior in $\mathbb{R}^{n}$

Can we define the concept of limsup and liminf in $\mathbb{R}^n$, the Euclidean $n$ space? Like for real valued function with domain in $\mathbb{R^n}$?
nini's user avatar
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1 vote
1 answer
75 views

Example of $\liminf$ and $\limsup$ of events

I am new to probability and I'm trying to understand the concept of $\liminf$ and $\limsup$ of events through an example. I am very comfortable with the technical definitions given by set operations ...
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