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.
1,894
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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\...
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3
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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+...
- 31
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1
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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 ...
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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 $...
- 151
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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_{\...
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1
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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
$$
s_1=0 ; \quad s_{2 m}=\frac{s_{2 m-1}}{2} ; \quad s_{2 ...
- 29
1
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2
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Compute $\limsup_{n\to \infty} |\sin n|^n$
$$
\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$.
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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 ...
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1
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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 ...
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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\...
- 23.2k
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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 ...
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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 ...
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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 $$ for every sequence $(x_n)_{n \in \mathbb{N}}$ ...
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1
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Proof for Statement regarding Borel Cantelli Lemma [closed]
I am struggeling to apply the Borel-Cantelli lamma to the following problem:
Let $ A, A_1, A_2,\dots \in F $ in the probability space $ (\Omega, F, \textit{P}$) with $ \sum_{n \in \mathbb{N}} P ( A_n ∩...
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Elementary Analysis_limsup and sequences [duplicate]
Prove that $limsup|S_n|^{1/n}$ $\le$ $limsup|S_{n+1}/S_n|$
Let $a$ = $limsup|S_n|^{1/n}$ and $L$ = $limsup|S_{n+1}/S_n|$
To prove that $a$ $\le$ $L$ it would suffice to show that
$a$ $\le$ $L_1$ for ...
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Definition of limit superior and limit inferior of real functions
In Wikipedia, the definition of limit superior and limit inferior of functions from metric spaces is:
Take a metric space $X$, a subspace $E$ contained in $X$, and a function $f:E\to \mathbb {R}$. ...
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Constructing a counterexample, improper Integrals
Assume that i have a Riemann integrable function $f$ defined on $\mathbb{R}$ and i know that $$\int_1^{\infty}f(x)dx$$ exists. How can i show that this doesnt necessarily imply $\liminf\limits_{x\to\...
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Does $\liminf$ of a set always exist?
I'm little confused about the whole concept of limits of sets. I'm working with the definition:
$$\liminf_{n\to\infty}(A_n) := \bigcup_{n=1}^{\infty}\bigcap_{k=n}^{\infty}(A_k).$$
As an example ...
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2
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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 ...
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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 $...
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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, ...
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$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\...
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(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 ...
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1
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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 ...
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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}...
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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 $$I_s:=\sup_{f\...
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Showing that $\mu(\limsup_{n\to\infty}f^{-n}(A)\setminus A) = 0$ if $\mu(A)=\mu(f^{-1}(A)),\mu(A\setminus f^{-1}(A))=0$ and $\mu$ is finite
Let $\mu$ be a probability measure over $X$, $f:X\to X$ a $\mu$ measurable mapping and suppose that $\mu$ preserves $f$ in the sense that $\forall S\subset X:\mu(S) = \mu(f^{-1}(S))$. Let $A\subset X$ ...
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If $\{\lambda_n\} \subset \Bbb R_{>0}$, then $\limsup_m \left(\frac{\lambda_{n_m + 1}}{\lambda_{n_m}} \right) \le \liminf_n \, (\lambda_n)^{1/n}$
This paper states Lemma $2.3$ without proof. I am trying to come up with a proof for the same.
Lemma $2.3$: If $\{\lambda_n\}_{n\in \mathbb N}$ is any sequence of positive numbers, then there exists ...
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Proof of Riemann's theorem of Rudin how to show that $\alpha$ and $\beta$ are the $\lim \sup$ and $\lim \inf$ - need help to end my proof
Here is the text of Rudin, Principles of Mathematical Analysis, chapter $3$, theorem $3.54$:
Let $\sum a_n$ be a conditionally convergent series. Suppose :
$$
-\infty \leq \alpha\leq \beta \leq +\...
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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 ...
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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 ...
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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 ...
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1
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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 ...
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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 ...
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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 ...
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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 ...
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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 $...
- 390
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$\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 $t_n\to +\infty$
$$\...
1
vote
2
answers
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show that $\lim\limits_{n\to \infty}\sup (-1)^{n}n=\infty$
How do i show formally that : $\lim\limits_{n\to \infty}\sup (-1)^{n}n=\infty$.
I know that if n is odd, then lim inf will be $-\infty$ and if n is even then lim sup will be $\infty$. However, i dont ...
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Exchanging limsup with liminf
I am working on a problem where I think I might be able to complete my argument if I can show the following relation.
\begin{align*}
\limsup_{k \rightarrow \infty} \liminf_{x' \rightarrow x} \frac{1}{...
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1
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Show that $\lim\limits_{n\to\infty}D_n$ exists if and only if $\lim\limits_{n\to\infty}E_n=\emptyset$
This is one of the exercises of Halmos's measure theory book.
I know it has been already treated but I've been asked to compute it by going via indicator functions.
Exercises goes as follows:
Let $(...
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1
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39
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Equivalent defintion of $\liminf_{s \to t} f(s)$?
Let $D \subset \mathbb{R}$ and $f: D \to \mathbb{R}$ and $t$ a limit point of $D$. Is the definition
$$L:=\liminf_{s \to t} f(s):= \lim_{\epsilon \to 0} (\inf \{ f(s)| s \in D \cap (t-\epsilon,t+\...
user1123845
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1
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50
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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 ...
- 13
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1
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94
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$\lim\inf x_n$ 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 ...
- 1,949
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0
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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 : $$\sum^\infty_{n=20}\frac{n}{2}(...
- 766
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2
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40
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Suppose $X_n \to X$ in $L^p$. Show that E$|X^p_r| \to E|X^p|$.
Suppose $X_n \to X$ in $L^p$. Show that E$|X^p_r| \to E|X^p|$.
The proof suggested the use of Minkowski's inequality in order to get that:
$$
[E|X^p|]^{1/p} \leqslant [E(|X_n - X|^p)]^{1/p} + [E(|X^...
- 583
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vote
1
answer
60
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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, ...
- 3,093
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votes
1
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254
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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 ...
- 1,301
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0
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21
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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{...
- 445
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0
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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 ...
