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.

Filter by
Sorted by
Tagged with
0 votes
3 answers
40 views

Let $a_n$ and $x_n,y_n\ge0$ be sequences such that $(x_na_n)$is Cesaro summable, $mx_n\le y_n\le Mx_n$ for some $m,M>0$ , $|x_na_n|\le1$ and $x_n\to0$

Let $(a_n),(x_n)$ and $(y_n)$ be sequences of real numbers with $x_n,y_n\ge0$, $mx_n\le y_n\le M x_n$ for some $m,M>0$, $|x_n a_n|\le1$, $x_n\to0$ and $\lim\limits_{N\to\infty}\frac{1}{N}\sum\...
DeltaEpsilon's user avatar
  • 1,060
2 votes
3 answers
93 views

Let $\{x_n\}$ be sequence of real numbers such that $\lim_{n\to\infty}({x_{n+1}}-{x_n})=5$. Then find $\lim_{n\to\infty}\frac{x_n}{n \log n}$. [duplicate]

I am unable to solve this problem. I tried using the definition of the limit of sequences. I took an example by considering a sequence $x_n= 5n$. Then I got that, the limit was zero. $x_{n+1}-x_n=(5n+...
Mallick's user avatar
  • 31
1 vote
1 answer
75 views

Show a $\liminf$ statement involving probabilities

Take a sequence of binary random variables $(Y_t)_t$ such that $$ (A) \quad \Pr\Big(\lim_{T\rightarrow +\infty} \frac{1}{T}\sum_{t=1}^T Y_t=\nu\Big)=1. $$ Consider another sequence of discrete random ...
TEX's user avatar
  • 58
0 votes
0 answers
33 views

Is my proof of convergence for a sum of random variables correct?

Let $ X_i \text{ for } i \in \mathbb N $ be a sequence of independent and identically distributed random variables, with $X \sim \mathscr N_{0,1}$ and $ \alpha>2 $. $ S_n := \sum_{i=1}^n X_i$ and $...
100xln2's user avatar
  • 151
2 votes
0 answers
37 views

Convergence of the perimeter of level sets

Suppose you have a sequence of $C^1$ functions $\{\phi_n\}_{n\in \mathbb{N}}$ defined on $\mathbb{R}^n$ that converges in $C^{1}_{\mathrm{loc}}$ to a function $\phi$, as $n \to +\infty$. By $C^1_{\...
totallyimmersed9's user avatar
1 vote
1 answer
83 views

Assumption in solution to Baby Rudin Exercise 3.4?

From Rudin's Principles of Mathematical Analysis. Problem 3.4: Find the upper and lower limits of the sequence $\left\{s_n\right\}$ defined by $$ s_1=0 ; \quad s_{2 m}=\frac{s_{2 m-1}}{2} ; \quad s_{2 ...
Incubu121's user avatar
1 vote
2 answers
139 views

Compute $\limsup_{n\to \infty} |\sin n|^n$

$$ \limsup_{n\to \infty} |\sin n|^n $$ Consider $n$ is an integer. Since $n$ is an integer, I know that $\sin n$ is never equal to $1$. But still it is unclear if the lim sup is equal to $1$ or $0$.
Sundaresan G's user avatar
0 votes
0 answers
27 views

Question about limsup and liminf of slope

Consider the function $g:[0,1] \to \mathbb{R}$ defined by $g(x)=1 - \sqrt{0.5-x}$ when $x \leq 0.5$ and $g(x)=1$ otherwise. What is $\lim_{h \to 0^{+}} \inf \{\frac{g(0.5+t)-g(0.5)}{t} : 0< \lvert ...
Donut's user avatar
  • 145
-1 votes
1 answer
20 views

limsup convergence and series with positive terms

Let $\{a_{n}\}$ be a sequence of real numbers. Construct an example for which $a_{n}\ge 0$ and $limsup_{n}na_{n}=\infty$ and yet $\sum_{n}a_{n}$ is convergent. I found and understood the answer on ...
maths and chess's user avatar
1 vote
1 answer
67 views

A Diagonalization argument for Double limsup and /or liminf,

Let $\{F(k,n)\,:\:, n\geq 1, m\geq1\}$ be a double sequence family of extended real numbers $\Bbb R\cup\{-\infty,\infty\}$. I would like to prove the following statements 1- There is $(n_k)_k$, $n_k\...
Guy Fsone's user avatar
  • 23.2k
2 votes
0 answers
57 views

Proof that the limit inferior is less than or equal to the limit superior of a sequence

I have thought of a proof that the limit inferior of a sequence is less than or equal to the limit superior of that sequence as part of Exercise 6.4.3 part (c) from Tao's book Analysis I Fourth ...
Vaskara_GRek_O's user avatar
2 votes
1 answer
41 views

If $\lim P(A_n \le B_n + C_n) = 1$ and $C_n = o(1)$, then $\limsup (P(A_n > u) - P(B_n > u)) \le 0, \forall u \in \mathbb{R}^+$

The problem statement is in the title. $A_n, B_n$ are random sequences, $C_n$ is deterministic. The statement itself is complicated because of various notations. But I think the proof relies on this ...
maskeran's user avatar
  • 325
0 votes
1 answer
51 views

Relation between continuous and discrete limsup and liminf

I know that, given a function $f:\mathbb{R} \to\mathbb{R}$ we have that: $$ \lim _{x \to x_0} f(x)=\lambda \iff \lim _{n \to +\infty} f(x_n)=\lambda $$ for every sequence $(x_n)_{n \in \mathbb{N}}$ ...
Luigi Traino's user avatar
-2 votes
1 answer
47 views

Proof for Statement regarding Borel Cantelli Lemma [closed]

I am struggeling to apply the Borel-Cantelli lamma to the following problem: Let $ A, A_1, A_2,\dots \in F $ in the probability space $ (\Omega, F, \textit{P}$) with $ \sum_{n \in \mathbb{N}} P ( A_n ∩...
Alligatooo's user avatar
0 votes
0 answers
18 views

Elementary Analysis_limsup and sequences [duplicate]

Prove that $limsup|S_n|^{1/n}$ $\le$ $limsup|S_{n+1}/S_n|$ Let $a$ = $limsup|S_n|^{1/n}$ and $L$ = $limsup|S_{n+1}/S_n|$ To prove that $a$ $\le$ $L$ it would suffice to show that $a$ $\le$ $L_1$ for ...
Vinny Aleksandrov's user avatar
0 votes
0 answers
31 views

Definition of limit superior and limit inferior of real functions

In Wikipedia, the definition of limit superior and limit inferior of functions from metric spaces is: Take a metric space $X$, a subspace $E$ contained in $X$, and a function $f:E\to \mathbb {R}$. ...
user136524's user avatar
2 votes
1 answer
123 views

Constructing a counterexample, improper Integrals

Assume that i have a Riemann integrable function $f$ defined on $\mathbb{R}$ and i know that $$\int_1^{\infty}f(x)dx$$ exists. How can i show that this doesnt necessarily imply $\liminf\limits_{x\to\...
Mathbds's user avatar
  • 59
1 vote
1 answer
113 views

Does $\liminf$ of a set always exist?

I'm little confused about the whole concept of limits of sets. I'm working with the definition: $$\liminf_{n\to\infty}(A_n) := \bigcup_{n=1}^{\infty}\bigcap_{k=n}^{\infty}(A_k).$$ As an example ...
Numik's user avatar
  • 13
1 vote
2 answers
70 views

Existence of limit of $f^2$ implies existence of limit of $f$

Consider a continuous function $f : \mathbb R \to \mathbb R$ such that $f(x)^2$ has a limit as $x\to\infty$. I am wondering if this implies the existence of the limit of $f(x)$ as $x\to\infty$. One ...
MSDG's user avatar
  • 7,081
0 votes
0 answers
15 views

Show that $\mathbb{I}_A$ is a lower semi continuous function for an open set $A$ in $\mathbb{R}$

Suppose $A$ is an open set in $\mathbb{R}$. Show that $\mathbb{I}_A$ is a lower semi continuous function where, a function, $f:\mathbb{R}\rightarrow\mathbb{R}$ is said to be lower semi continuous at $...
zaira's user avatar
  • 1,949
0 votes
0 answers
40 views

Proving the definition of the limit superior of a set S for an infinite bounded set S.

So I'm self studying from Schramm's Introduction to Real Analysis book, and I am currently at Chapter 7, on cluster points. I thought that I could use any input regarding improving my technique, ...
Lightwalker's user avatar
0 votes
2 answers
59 views

$F$ is lower semicontinuous $\iff F(x)=\sup_{r>0}\inf_{y\in B(x,r)}F(y)$ for all $x\in X$

Let (X,d) be a metric space. $F: X\to \overline{\mathbb{R}}$. The definitions that I have to use are: (1) $F$ is sequentially lower semicontinuous if for all sequences $(x_n)_n \subseteq X$ s.t $ x_n\...
some_math_guy's user avatar
0 votes
1 answer
38 views

(X,d) be a metric space. $F: X\to \overline{\mathbb{R}}$. $F$ is sequentially lower semicontinuous iff $F$ is lower semicontinuous

Let (X,d) be a metric space. $F: X\to \overline{\mathbb{R}}$. $F$ is sequentially lower semicontinuous iff $F$ is lower semicontinuous The definitions that I have to use are: (1) $F$ is sequentially ...
some_math_guy's user avatar
1 vote
1 answer
102 views

How do I prove these results involving the liminf and the inf ? ( part of a proof on sequential lower semicontinuity of lower semicontinuous envelope)

I am trying to prove equation (6.3) in the lemma below. This is part of a course in calculus of variations, but that is irrelevant here, this is actually a question about the liminf and the inf of ...
some_math_guy's user avatar
1 vote
1 answer
44 views

Supremum of limit inferior

Define the limit inferior of a sequence of sets as $$\liminf A_n = \bigcup_{N \geq 0} \bigcap_{n \geq N} A_n , $$ and the limit inferior of a sequence as $$\liminf X_n = \sup_{N \geq 0} \inf_{n \geq N}...
Marm's user avatar
  • 11
3 votes
0 answers
33 views

Hints on proving $\liminf_{s\to 0}\sup_{f\in L^2(X):||f||_{L^2}\leq 1}\int_X|f(x)|^2|e^{-is/||x||}-1|^2dx>0$ for $X=\mathbb{R}^n$

Take $X = \mathbb{R}^n$, denote by $\|x\|$ the standard Euclidean norm and consider the space of $L^2$ functions over $X$ in the Lebesgue measure. I am trying to show that the sequence $$I_s:=\sup_{f\...
Cartesian Bear's user avatar
0 votes
1 answer
29 views

Showing that $\mu(\limsup_{n\to\infty}f^{-n}(A)\setminus A) = 0$ if $\mu(A)=\mu(f^{-1}(A)),\mu(A\setminus f^{-1}(A))=0$ and $\mu$ is finite

Let $\mu$ be a probability measure over $X$, $f:X\to X$ a $\mu$ measurable mapping and suppose that $\mu$ preserves $f$ in the sense that $\forall S\subset X:\mu(S) = \mu(f^{-1}(S))$. Let $A\subset X$ ...
Epsilon Away's user avatar
  • 1,038
2 votes
2 answers
126 views

If $\{\lambda_n\} \subset \Bbb R_{>0}$, then $\limsup_m \left(\frac{\lambda_{n_m + 1}}{\lambda_{n_m}} \right) \le \liminf_n \, (\lambda_n)^{1/n}$

This paper states Lemma $2.3$ without proof. I am trying to come up with a proof for the same. Lemma $2.3$: If $\{\lambda_n\}_{n\in \mathbb N}$ is any sequence of positive numbers, then there exists ...
stoic-santiago's user avatar
2 votes
1 answer
76 views

Proof of Riemann's theorem of Rudin how to show that $\alpha$ and $\beta$ are the $\lim \sup$ and $\lim \inf$ - need help to end my proof

Here is the text of Rudin, Principles of Mathematical Analysis, chapter $3$, theorem $3.54$: Let $\sum a_n$ be a conditionally convergent series. Suppose : $$ -\infty \leq \alpha\leq \beta \leq +\...
niobium's user avatar
  • 969
0 votes
1 answer
52 views

Series with $\lim \sup$ and $\lim \inf$ being $+$ and $-\infty$

Do you have an example of a series whose partial sums' $\lim \sup = \infty$ and whose partial sums' $\lim \inf = -\infty$ ? I feel like it's like repeatingly adding a whole bunch of positive terms ...
niobium's user avatar
  • 969
0 votes
2 answers
92 views

How to prove that $L^- \le c \le L^+$. Where $c$ is a limit point and $L^-$ and $L^+$ are the limit inferior and limit superior respectively.

I have been trying to prove this exercise for way too long and am nowhere near a proof. I have asked this question before and haven't got a satisfactory answer. This could be my own fault for not ...
Seeker's user avatar
  • 3,504
0 votes
1 answer
58 views

Prove that $\varliminf_{n \to \infty}\left(\inf_{x\in X} f_n(x)\right) \leq \inf_{x\in X}\left( \varliminf_{n \to \infty} f_n (x)\right)$

We also know that $f_n(x)$ is bounded for every $n\in\mathbb{N}, x\in X$. Firstly using liminf definition I rewrote $\varliminf_{n \to \infty} f_n(x) = \lim_{n \to \infty} \inf_{k\geq n} f_k(x)$ And ...
pollumees's user avatar
1 vote
1 answer
69 views

Inequality involving changing order of limits and probability

I read this paper, in Corollary 1 the author claims that $$\underset{\pi \in [0, 1]}{\sup}\ W_T(\pi) \overset{p}{\to} \infty$$ as $T \to \infty$. Where $W_T(\pi)$ is Wald statistics but I think it ...
D F's user avatar
  • 1,331
1 vote
1 answer
43 views

A question on upper and lower derivatives of $F$ on $[a,b]$

This question is from the book The Integrals of Lebesgue, Denjoy, Perron, and Henstock. I'm currently reading part of the book The Integrals of Lebesgue, Denjoy, Perron, and Henstock. Slight ...
raijin's user avatar
  • 155
5 votes
1 answer
68 views

Limit inferior of bounded sequence [duplicate]

I found an interesting problem that I can't tackle as I am studying real analysis on my own. Let there be bounded sequences $(a_n)$ and $(b_n)$. Proove that $\varliminf_{n\to\infty} (\min \{ a_n , b_n ...
Amanda S.'s user avatar
0 votes
0 answers
23 views

Prove $F(x-)\le\lim\inf F_n(x-)$ if $F_n\overset{d}{\rightarrow}F$

Let $F_n\overset{d}{\longrightarrow}F$, then $\forall x\in\mathbb{R}$. Then, $$F(x-)\le\lim\inf F_n(x-)\le\lim\sup F_n(x)\le F(x)$$ I have the following: Suppose $x_n\rightarrow x\in\mathbb{R}$. For ...
zaira's user avatar
  • 1,949
0 votes
1 answer
36 views

Pointwise limit and limsup of sets

I don't think this problem should be too difficult, but I'm not sure on some details. Give a sequence of functions $f_{n} : \mathbb{R} \rightarrow \mathbb{R}$ converging for any $x \in \mathbb{R}$ to $...
beeselmane's user avatar
1 vote
0 answers
24 views

$\limsup$ of functions and sequential $\limsup$.

Let $f:(0,+\infty) \to \mathbb{R}$, a continous and non negative function and $g:(0,1) \to \mathbb{R}$ a non negativa continuous function, if for all sequence $(t_n)$ such that $t_n\to +\infty$ $$\...
Jarbas Dantas Silva's user avatar
1 vote
2 answers
71 views

show that $\lim\limits_{n\to \infty}\sup (-1)^{n}n=\infty$

How do i show formally that : $\lim\limits_{n\to \infty}\sup (-1)^{n}n=\infty$. I know that if n is odd, then lim inf will be $-\infty$ and if n is even then lim sup will be $\infty$. However, i dont ...
user20194358's user avatar
0 votes
0 answers
42 views

Exchanging limsup with liminf

I am working on a problem where I think I might be able to complete my argument if I can show the following relation. \begin{align*} \limsup_{k \rightarrow \infty} \liminf_{x' \rightarrow x} \frac{1}{...
GA-Student's user avatar
5 votes
1 answer
136 views

Show that $\lim\limits_{n\to\infty}D_n$ exists if and only if $\lim\limits_{n\to\infty}E_n=\emptyset$

This is one of the exercises of Halmos's measure theory book. I know it has been already treated but I've been asked to compute it by going via indicator functions. Exercises goes as follows: Let $(...
Ricter's user avatar
  • 583
0 votes
1 answer
39 views

Equivalent defintion of $\liminf_{s \to t} f(s)$?

Let $D \subset \mathbb{R}$ and $f: D \to \mathbb{R}$ and $t$ a limit point of $D$. Is the definition $$L:=\liminf_{s \to t} f(s):= \lim_{\epsilon \to 0} (\inf \{ f(s)| s \in D \cap (t-\epsilon,t+\...
user avatar
0 votes
1 answer
50 views

Swap $\limsup$ and $\mathbb{E}$ of sequence of functions evaluated at a random variable?

I am a postgrad with more of a background in the functional analysis point of view of things but am recently needing to get the expectation involved and it has been a few years since I've done much ...
Alphatronic's user avatar
0 votes
1 answer
94 views

$\lim\inf x_n$ can possibly be $-\infty$ or $\infty$? Either or Both?

Show that $\lim\inf x_n$ always exists for any sequence $x_n$ of real numbers and can possibly be $-\infty$. Proof for the existence part is easy if I assume the sequence is bounded finitely. It ...
zaira's user avatar
  • 1,949
1 vote
0 answers
22 views

Radius of convergence of complex series $\sum^\infty_{n=10}\frac{n}{2}(z-1)^{n}$ [duplicate]

I'm trying to find the radius of convergence of the following complex series: $$\sum^\infty_{n=10}n(z-1)^{2n}$$ I've started by rewriting the above series as follows : $$\sum^\infty_{n=20}\frac{n}{2}(...
John Katsantas's user avatar
0 votes
2 answers
40 views

Suppose $X_n \to X$ in $L^p$. Show that E$|X^p_r| \to E|X^p|$.

Suppose $X_n \to X$ in $L^p$. Show that E$|X^p_r| \to E|X^p|$. The proof suggested the use of Minkowski's inequality in order to get that: $$ [E|X^p|]^{1/p} \leqslant [E(|X_n - X|^p)]^{1/p} + [E(|X^...
Ricter's user avatar
  • 583
1 vote
1 answer
60 views

The liminf and limsup in terms of accumulation points

Let $(x_n)_{n\in\mathbb N}$ be a sequence of real numbers. If it is bounded, then the Bolzano–Weierstrass theorem tells us that the set of accumulation points in $\mathbb R$ is non-empty. Furthermore, ...
Filippo's user avatar
  • 3,093
6 votes
1 answer
254 views

Optimal bounds for the product of the divisor function $d(n)$ in short intervals

Let $d(n)$ denote the number of divisors of a positive integer $n$. It is pretty obvious that $d(n) \ge 2$ for any given number $n \ge 2$, since every number is divisible by $1$ and itself. $2$ is ...
Kinheadpump's user avatar
  • 1,301
0 votes
0 answers
21 views

Subdifferential and superdifferential with respect to $x$

Let $\Omega \subseteq \mathbb{R}^n$ be an open set and let $v:\Omega \to \mathbb{R}$ be a continuous function. The subdifferential of $v$ at $x \in Q$ is the set $D^-v(x)=\{p \in \mathbb{R} \mid \text{...
effezeta's user avatar
  • 445
1 vote
0 answers
36 views

Mill's ratio and convergence almost surely

Let ${X_n}$ be a sequence of independent random variables N[0,1]. Show that: $$ \mathcal{P}(\underbrace{lim sup}_{n \longmapsto \infty} \frac{|X_n|}{\sqrt{log \ n}} = \sqrt{2}) = 1 $$ I've been asked ...
Hinata Takahashi's user avatar

1
2 3 4 5
38