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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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1answer
25 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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0answers
15 views

limsup of subsequences [on hold]

For a bounded real sequence $(x_n)$ let $\limsup x_n=L$. If $t>1$ it is true that $\limsup x_{[t_n]}=L$ where $[t_n]$ denotes the integer part of the product $tn$?
3
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2answers
59 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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0answers
50 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
32 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
32 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
66 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
60 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
50 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
114 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
44 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
30 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 ...
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0answers
97 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
26 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 ...
2
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1answer
47 views

A property of limit supremum

I am just reading about limit supremum concept in Rudin. Rudin defines the limit supremum as follows: Let $\{a_n\}$ be a sequence in $[\infty,\infty]$. $b_k = \sup \{a_k, a_{k + 1}, \ldots, \}, \...
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0answers
15 views

limsup of a function related to a bounded succession

Given $f: \mathbb R \to \mathbb R$ a function with this property: for every real and bounded sequence $a_n$ $$ f(\limsup\limits_{n\to\infty} a_n) = \limsup\limits_{n\to\infty}f(a_n)$$ Prove that $f$ ...
1
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1answer
46 views

(Proof verification) Showing that the two $\limsup$ definitions are equivalent

I have been trying to prove that the two definitions of $\limsup$ are equivalent. I would appreciate it if someone could verify my attempt! Thanks in advance! Here are the two definitions: ...
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5answers
699 views

Stuck at proving whether the sequence is convergent or not

I have been trying to determine whether the following sequence is convergent or not. This is what I got: Exercise 1: Find the $\min,\max,\sup,\inf, \liminf,\limsup$ and determine whether the ...
2
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1answer
29 views

$\limsup_{n \to \infty}\{\frac{X_{n}}{\log{(n)}}\leq\epsilon\}\subseteq \{\liminf_{n \to \infty}\frac{X_{n}}{\log{(n)}}\leq\epsilon\}$

I recently saw an assertion made that $$\bigcap_{m \in \mathbb N} \bigcup_{n \geq m}\left\{\frac{X_{n}}{\log{(n)}}\leq\epsilon\right\} \subseteq\left\{\liminf_{n \to \infty}\frac{X_{n}}{\log{(n)}}\...
1
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1answer
37 views

Limit inferior of product of two non negative bounded sequences

I have two contradicting proofs for the property: \begin{align}\liminf (a_nb_n) \geq (\liminf (a_n)) * (\liminf (b_n))\end{align} where ${a_n}$ and ${b_n}$ are two bounded non-negative sequences. (...
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2answers
56 views

Radius of convergence of power series $\sum_{n=0}^{\infty} n!x^{n^2}$

Radius of convergence of power series $\sum_{n=0}^{\infty} n!x^{n^2}$ $\sum_{n=0}^{\infty} n!x^{n^2} = 1 + x + 2x^4 + 6x^9\ldots$ Comparing this with $\sum_{n=0}^{\infty} a_nx^n=$ $a_n= n! $ or $0$ ...
1
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2answers
36 views

Silly question on lower semi-continuity

Suppose that $X_n\rightarrow X$ in a complete separable metric space $(\mathcal{X},d)$. Let $f:\mathcal{X}\rightarrow (-\infty,\infty]$ be a proper, convex, lower semi-continuous function, such that $...
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0answers
26 views

Calculation of $\liminf_{n\to\infty} \frac{1}{1+e^{nx}}e^{-x^2/n} $ on $\mathbb{R}^-$

I would intuitively suggest that it is something like: $1$ for $x<0$ and $0.5$ for $x=0$, what do you think?
3
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0answers
29 views

Show that the distance $d(x)$ of a point $x = (x_k)\in \ell_\infty$ to $c_0$ is equal to $\limsup_{k\to\infty} |x_k|.$

Question: Show that the distance $d(x)$ of a point $x = (x_k)\in \ell_\infty$ to $c_0$ is equal to $\limsup_{k\to\infty} |x_k|.$ Thus the norm in $\ell_\infty / c_0$ is $\|\hat{x}\| = \...
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2answers
33 views

$\limsup_{n\to\infty}{a_n}^{b_n}= \limsup_{n\to\infty}{a_n}^{\lim\limits_{n\to\infty}b_n}$ under certian conditions.

I have a proof that requires the following justification. So, I decided to recast it in the following form: If $\lim\limits_{n\to\infty}b_n$ exists and $\limsup_{n\to\infty}{a_n}=a>0$, then $\...
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3answers
50 views

If $\lim \text{sup}_{n\to\infty}X_n≠\lim \text{inf}_{n\to\infty}X_n$, can we say “limit doesn't exist”?

According to the definition of the limit, if $\lim \text{sup}_{n\to\infty}X_n=A$ and $\lim \text{inf}_{n\to\infty}X_n=B$ where, $A≠B$. Terminologically, can we say "limit doesn't exist"? Exact ...
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0answers
59 views

Are these limits correct? $\lim_{n\to \infty}\text{sup} \frac{\lambda (n)}{n}=1$ and $\lim_{n\to \infty}\text{inf} \frac{\lambda (n)}{n}=0$ exist?

I learned that from here for Euler totient function $\phi (n)$ , we have $$\lim_{n\to \infty}\text{sup} \frac{\phi (n)}{n}=1$$ $$\lim_{n\to \infty}\text{inf} \frac{\phi (n)}{n}=0$$ However, ...
1
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0answers
55 views

Infimum of sum of two values.

Let the function $f_{n}:\mathbb{R}\rightarrow \mathbb{R_{\ge 0}}$ be defined as $$ f_{n}(x)=\sum_{i,j=1}^{n}\frac{(-1)^{i+j}\cos(\ln \frac{i}{j})}{(ij)^{x}}\quad \forall n\in\Bbb N $$ There is given ...
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0answers
166 views

Complex Analysis Extension of Bernoulli Number Generating function

I have just recently started revising for my complex analysis module at university and come across and interesting exercise in a textbook while reading. I vaguely understand the concept but am ...
3
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0answers
82 views

About “Principles of Mathematical Analysis” by Walter Rudin Theorem 3.17(a).

I am reading Walter Rudin's "Principles of Mathematical Analysis". There are the following definition and theorem and its proof in this book. Definition 3.16: Let $\{ s_n \}$ be a sequence ...
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0answers
40 views

$\limsup$ and integral inequality

I have functions $f$, $g$ :$(0, \infty) \rightarrow (0, \infty)$, and $g$ is invertible and decreasing (I don't know if it is relevant or not in this case). I know also that $\limsup_{x \searrow 0} \...
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2answers
86 views

Walter Rudin “Principles of Mathematical Analysis” Definition 3.16, Theorem 3.17. I cannot understand.

I am reading Walter Rudin's "Principles of Mathematical Analysis". There are the following definition and theorem and its proof in this book. Rudin didn't prove that $E \neq \emptyset$. Why? ...
3
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1answer
53 views

Stolz-Cesàro $0/0$ case: is $\limsup \frac{a_n}{b_n}\le \limsup\frac{a_{n+1}-a_n}{b_{n+1}-b_n}$?

The general form of Stolz-Cesaro $\infty/\infty$ case states that any two real two sequences $a_n$ and $b_n$, with the latter being monotone and unbounded, satisfy $$\liminf\frac{a_{n+1}-a_n}{b_{n+1}-...
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0answers
38 views

Let $f$ be Lebesgueable integrable, then $\lim_{h\to 0}\sup \int_{|x -y| < h}|f(x) - f(y)| = 0$

I must show that Let $f$ be Lebesgueable integrable, then $$\lim_{h\to 0}\sup \int_{|x -y| < h}|f(x) - f(y)|dy = 0$$ for almost every $x \in \mathbb{R}$. Tentative proof: $$|f(x) - f(y)| \geq ...
2
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1answer
58 views

If $\{a_n\}$ is bounded and non-decreasing, prove that $\liminf b_n = 0$, $b_n = n(a_{n+1} - a_{n})$

Let $\{a_n\}$ be a bounded and non-decreasing sequence of reals, and $b_n = n(a_{n+1} - a_{n})$. (a) Show that $\liminf b_n = 0$ (b) Give an example of a sequence $\{a_n\}$ such that $\{b_n\}$ ...
1
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1answer
39 views

Intuition behind inequality for measure of $\liminf$ and $\limsup$

For a set $X$ with a $\sigma$-algebra $\xi \subseteq \mathcal{P}(X)$ and $\sigma$-additive $\mu: \xi \rightarrow [0, \infty]$. The following inequality holds for $(A_n)_{n \in \mathbb{N}} \in \xi^\...
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1answer
52 views

$\liminf E_{n_0+n} = \liminf\limits E_{n}$ and $\limsup E_{n_0+n} = \limsup E_{n}$ where $E_n$ is a decreasing sequence?

I dropped the $n\rightarrow\infty$ in the title as it was exceeding the character limit. In the book I'm currently reading, the author claims that $\liminf\limits_{n\rightarrow\infty} E_{n_0+n} = \...
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0answers
46 views

For a convergent sequence $\limsup b_n = \lim b_n$

Happy new year! I wish to prove the following result to practice with $\limsup: $ For a convergent sequence$ (b_n)$ we know $\limsup b_n = \lim b_n$ If $b_n$ is convergent, the set of ...
4
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2answers
47 views

Finding $\limsup_{n\to\infty}\sum_{k=0}^n(-1)^k|\sin k|$

How can we find $$L=\limsup_{n\to\infty}\sum_{k=0}^n(-1)^k|\sin k|,$$ where $|\cdot|$ denotes the absolute value of $\cdot$? According to this answer, we can see the limit does exists. ...
1
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1answer
53 views

Is $\limsup_{n\rightarrow\infty}\sqrt[n]{|a_n|}=\frac{1}{R}=\limsup_{n\rightarrow\infty}|\frac{a_{n+1}}{a_n}|$?

$R$ is the radius of convergence for a powerseries I will write down my proof but I am not sure whether this is right because I thought $$\limsup_{n\rightarrow\infty}\sqrt[n]{|a_n|}\leq \limsup_{n\...
1
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2answers
36 views

Clarification on step in proof for Ratio test

My thoughts: So apart from this step, the rest of the proof is fairly simple. Now in terms of lim sup, I know that it is just equal to the regular limit for convergent sequences and for bounded, ...
3
votes
4answers
60 views

Prove any number $c \in [a, b]$ is a subsequential limit if $\lim\inf x_n = a$, $\lim \sup x_n = b$, $a\ne b$, $\lim(x_n -x_{n+1})=0$

I'm trying to solve the following problem: Let $\{x_n\}$ denote a bounded sequence. Prove that any number $c \in [a, b]$ is a subsequential limit of $\{x_n\}$ if: $$ \begin{cases} \lim_{n\to\...
1
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2answers
58 views

Existence of a subsequence converging to limsup

Let $(a_n)$ be a bounded sequence of real numbers, and define $$\beta_n = \sup \{ a_k : k \geq n \}. $$ This sequence converges to a limit, $$\lim_{n \to \infty} \beta_n = \limsup a_n.$$ I'm ...
2
votes
1answer
61 views

Proof verification for $\lim \inf x_n + \lim \sup y_n \le \lim \sup(x_n + y_n)$ for bounded $x_n, y_n$

Let $\{x_n\}$ and $\{y_n\}$ denote two bounded sequences. Prove that: $$ \lim \inf x_n + \lim \sup y_n \le \lim \sup(x_n + y_n) \\ $$ We know that both $x_n$ and $y_n$ are bounded hence is their ...
1
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2answers
37 views

Lim Sup of n-th root of integrable functions

I am working on some analysis problems for a qualifying exam and came across this one which has given me some problems: Let $f_{n}$ be $\textbf{nonnegative}$ measurable functions on a measure space $(...
1
vote
1answer
47 views

Show that $\limsup_{k \to \infty} 2^{-k} N_k = 0$ where $N_k$ is the number of $a_n \geq 2^{-k}$.

Let $\{a_n\}_{n \geq 1}$ be a non-negative sequence of reals such that $\sum_{n \geq 1} a_n$ converges to $s$. Define $N_k = |\{n \in \mathbb{N} : a_n \geq 2^{-k}\}|$. Show that \begin{equation} \...
1
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1answer
33 views

Does $\liminf$ distribute?

I know the property that for sets $A_n,B_n$ $$ \limsup_{n\to\infty}(A_n\cup B_n)=\limsup_{n\to\infty}(A_n)\cup \limsup_{n\to\infty}(B_n) $$ holds. I'm curious if it also holds that $$ \liminf_{n\to\...
6
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0answers
53 views

Solution verification: evaluate $\lim\limits_{n \to \infty}\frac{1^{\lambda n}+2^{\lambda n}+\cdots+n^{\lambda n}}{n^{\lambda n}}$ where $\lambda>0.$

Problem Evaluate $$\lim_{n \to \infty}\frac{1^{\lambda n}+2^{\lambda n}+\cdots+n^{\lambda n}}{n^{\lambda n}}$$ where $\lambda>0.$ Solution Denote $$S_n:=\sum_{k=1}^{n}\left(\frac{k}{n}\right)^{\...
0
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1answer
54 views

How can $\limsup_{x \to x_0} f(x) = f(x_0)$ for $f$ discontinuous at $x_0$?

My textbook says $$\limsup_{x \to x_0} f(x) = \max\{f(x_0), \lim_{h\to 0^+} f( x_0 + h), \lim_{h\to 0^-} f( x_0 + h)\}$$ Assuming $f(x_0)$ is distinct from the latter two values, how can $\limsup f(...
0
votes
0answers
33 views

Prove limit and liminf of series

I am self-studying convergence of sequence and series and have some hard time doing the following problem: Suppose $\sum_{k=0}^{\infty} a_kb_k^2 < \infty$. (a) If $\sum_{k=0}^{\infty}a_k = \infty$ ...