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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21 views

if $ \limsup_n x_n\leq y $then there exists a subsequence $\{x_{n_i}\}_i$ such that, for all $i\in\mathbb{N}$: $ x_{n_i}\leq y $? [closed]

Let $\{x_n\}_n$ be a real sequence and $y\in\mathbb{R}$ such that: $$ \limsup_n x_n\leq y $$ can we say that there exists a subsequence $\{x_{n_i}\}_i$ such that, for all $i\in\mathbb{N}$: $$ x_{n_i}\...
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3answers
47 views

I don't understand $\limsup_n\:A_n$ in Probability Theory, why should it be equal to “infinitely many $A_n$ occur”?

The definition for limsup is: $$\bigcap _{n=1}^{\infty }\bigcup _{k=n}^{\infty }\:A_k$$ So this means that $$\left(A_1\cup \:A_2\cup \:A_3\cup \:...\right)\cap \left(A_2\cup \:\:A_3\cup \:\:...\...
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0answers
35 views

An equivalent definition for the limsup $a_n$

Suppose that for $(a_n)$ the limit superior is finite. Prove the following statement: $$ L = \limsup_{n \to \infty} a_n \iff [ \forall \varepsilon>0 \exists k \in \mathbb{N} : \forall n >...
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44 views

What is a correct definition of liminf? [duplicate]

I heard about two definitions of liminf to sequences: The first one Consider sequence $a=\{x_1,x_2,x_3,...\}$ and slice it from some some index k via obtaining $a_k=\{a_k, a_{k+1},...\}$ subsequence....
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1answer
38 views

Covergence of sequence of sum of M Cosines

I have been working on the original problem and have narrowed it down to proving that : $\displaystyle lim_{n \to \infty} \sum_{i=1}^{M}\left(k_{i}\cos\left(n\theta_{i}-d_{i}\right)\right)$ does not ...
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1answer
48 views

Why is strict inequality needed here to prove a quantity the supremum? (easy analysis)

EDIT: answer provided. Just a slip up confusing the order of taking $sup$. EDIT to EDIT: While I did make the conceptual mistake above, turns out my conclusion was not incorrect. I have a function $...
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1answer
33 views

A question on limit sup

When is it the case that $\lim \sup _{k}\left(\alpha a_{k}\right)=\alpha \lim \sup _{k} a_{k} ?$ I am trying to help my friend.
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1answer
55 views

A question on limitsup

What is the relationship between $\lim \sup _{n}\left(a_{n} b_{n}\right)$ and $(\lim \sup _{n} a_{n})(\lim \sup _{n} b_{n})$?
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1answer
15 views

A doubt on the proof of Martingale Convergence Theorem on Jacod-Protter

Theorem: Let $(X_n)_{n\geq1}$ be a submartingale such that $\sup\limits_{n}\mathbb{E}\{X_n^{+}\}<\infty$. Then, $\lim\limits_{n\rightarrow\infty} X_n = X$ exists a.s. (and is finite a.s.). Moreover,...
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1answer
42 views

Different definitions for $\limsup$

I am wondering if the following various definitions of $\limsup$ are equivalent: $$\displaystyle{\limsup_{n \to \infty} x_n = \lim_{n \to \infty} \sup_{m \geq n} \{ x_m \}}$$ $$\displaystyle{\...
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0answers
34 views

Doubt on limit of a sequence of random variables

Let's define a probability space $(\Omega$, $\mathcal{F}$, $\mathbb{P})$ and a sequence of random variables $(X_n)_{n\geq0}$ defined on it. If I assume that the limit of such a sequence of random ...
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0answers
26 views

$E[|X|^r]=+\infty$ $\implies$ $P(\left\{\limsup_nY_n=+\infty \right\}\cup\left\{\liminf_nY_n=-\infty \right\})=1?!$

Let $(X_n)_n$ be a sequence of independent and identically distributed random variable, such that there exists $r>0$ such that $E[|X_1|^r]=+\infty.$ Let $(x_n)_n$ be a sequence of real numbers, $...
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1answer
18 views

Limit superior of a fraction involving two hyperharmonic series

I am trying to show that for $p\geq1$ $$\limsup_{k\to\infty}\frac{\sqrt{k}\sum_{n=k}^\infty n^{-3p}}{\left(\sum_{n=k}^\infty n^{-2p}\right)^{3/2}}<\infty.$$ Some numerical calculations suggest that ...
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0answers
30 views

Finding $\liminf\limits_{ r \to 1^{-}}$ and $\limsup\limits_{ r \to 1^{-}} \sum\limits_{k=0}^\infty (-1)^k r^{2^k}$

In an answer last year to another question, I implicitly asserted as an aside that that $\lim\limits_{r \to 1^-} \sum\limits_{k=0}^{\infty} (-1)^k r^{2^k}$ seemed empirically to be close to $\frac12$. ...
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29 views

liminf and $\max$

Let $u_n$ and $v_n$ two sequences of real numbers. It's true , if $(u_n)$ converges to finite limit L, that $\displaystyle \quad \liminf_n \max(u_n,v_n)=\liminf_n \max(L,v_n)$ ? if not, what we ...
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1answer
33 views

calculating $\limsup_{n \to \infty} \left(1+\frac{(-1)^n-3}{n} \right)^{n}$

I was asked in an exercise to check whether a series converges or not and while doing so I got the following limit to solve: $$\limsup_{n \to \infty} \left(1+\frac{(-1)^n-3}{n} \right)^{n}$$ I did ...
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2answers
37 views

limit superior of alternating series

I was asked in an exercise to check whether the following series converges or not: $$\sum_{n=1}^{\infty}\left(1+\frac{(-1)^n-3}{n} \right)^{n^2}$$ I used the fact that if a series $(a_n)_n$ converges,...
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1answer
31 views

If $c$ is any limit point of $(a_{n})_{n=m}^{\infty}$, then we have that $L^{-}\leq c\leq L^{+}$.

Let $(a_{n})_{n=m}^{\infty}$ be a sequence of real numbers, let $L^{+}$ be the limit superior of this sequence, and let $L^{-}$ be the limit inferior of this sequence (thus both $L^{+}$ and $L^{-}$ ...
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0answers
17 views

Prove that for every $x > L^{+}$, there exists an $N\geq m$ such that $a_{n} < x$ for all $n\geq N$.

Let $(a_{n})_{n=m}^{\infty}$ be a sequence of real numbers, let $L^{+}$ be the limit superior of this sequence, and let $L^{-}$ be the limit inferior of this sequence (thus both $L^{+}$ and $L^{-}$ ...
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1answer
44 views

Tail random variable and extremum

Let $(X_n)_n$ be a sequence of real random variables. Is it true that $\limsup_n\frac{1}{n}\max_{1 \leq k\leq n}X_k$ is tail random variable? $\liminf_n \frac{1}{n}\min_{1 \leq k \leq n}X_k$ ? $\...
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1answer
41 views

Bounded Dini derivative

I'm so stuck right now. I feel like I lost all my analysis skills. Assume I have a continuous map $s:[0,\infty) \to [0,\infty)$ with bounded Dini derivative for all $t_0 > 0$, in particular $$\...
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0answers
24 views

Exchange of limit and limsup for bounded and integrable function

Let $g$ be an integrable function such that $g : \mathbb{R}^+ \rightarrow \mathbb{R}^+$ and $g(x) < M$ for all $x$. Let $f_n: \mathbb{Z}^+ \rightarrow \mathbb{R}^+$ such that $$f_n(x) = \sum_{i=0}...
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0answers
44 views

Prove almost surely that a series is unbounded

Let $\{X_n\}_{n\ge1}$ be a sequence of independent and identically distributed random variables. Let $S_n = \sum_{n=1}^NX_n$ for $N\ge1$. Now assume that $X_n$ takes values $1, 0 \text{ or} -1$ with ...
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0answers
57 views

If $a_n\leq b_n$ then $\limsup a_n\leq \liminf b_n$

If $a_n\leq b_n$ then $\limsup a_n\leq \liminf b_n$ This is a statement in Royden, but I am not sure how to prove it.
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1answer
58 views

Do the lim sup and lim inf always exist?

Let's focus on the $\lim \inf$. While trying to prove that the $\lim \inf$ always exists, [this page]https://mathcs.org/analysis/reals/numseq/proofs/lub_ex.html established that the sequence $A_j = \...
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2answers
29 views

limsup and liminf of $a_n=\frac{n}{10^{\lceil log_{10}n \rceil} } $

For each natural number $n\ge 1$, let $a_n=\frac{n}{10^{\lceil log_{10}n \rceil} } $ , where $\lceil x \rceil=$ smallest integer greater than or equal to $x$ . Which of the following statements are ...
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1answer
51 views

How to prove $\limsup\limits{} \left(A_n \cap B_n\right)\subseteq\limsup\limits{} \left(A_n \right) \cap \limsup\limits{}\left(B_n \right) $? [duplicate]

Prove that for any collections $\{A_n\}$ and $\{B_n\}$, $$\limsup\limits{} \left(A_n \cap B_n\right)\subseteq\limsup\limits{} \left(A_n \right) \cap \limsup\limits{}\left(B_n \right) $$ Any hints ...
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0answers
22 views

Find lim sup and inf

you are given the two following sequences $a_n=\sqrt[n]{2}$ $b_n = \begin{cases} n & \text{if $n=even$} \\ 1/n & \...
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0answers
44 views

Is it trivial that $\limsup\limits_{n\to\infty}\sqrt[n]{|a_{n+1}|}=\limsup\limits_{n\to\infty}\sqrt[n]{|a_{n}|}$

Is it correct that $\underset{n\to\infty}{\limsup_{n\to\infty}\sqrt[n]{|a_{n+1}|}=\limsup}\sqrt[n]{|a_{n}|}$? i know its correct for regular $ \lim $ but im not sure for $ \limsup $. Also, is it ...
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1answer
21 views

Prove the existence of two constant to bound a function

Let $f:\mathbb{R}\longrightarrow\mathbb{R}$ a continous function, such as exists $p\in\mathbb{R}^+$, $p>1$ verifying: $$\liminf_{x\to\infty}\frac{f(x)}{|x|^p}=L\in(0,+\infty]$$ Prove the existence ...
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0answers
23 views

Inequality of limsup

For two complex sequences $(a_n)_n, (b_n)_n$ with $\limsup_{n \to \infty} |a_n|^{1/n} \geq \limsup_{n \to \infty} |b_n|^{1/n}$. I would like to show that $\limsup_{n \to \infty} |a_n + b_n|^{1/n} \...
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1answer
17 views

Prove the root test for sequence convergence with $\limsup$.

I am trying to prove the root test for sequence convergence with $\limsup$. I know exactly what to do but I think I misunderstood a simple detail when I learned about $\limsup$ and $\liminf$. That if ...
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1answer
24 views

Limit superior is a cluster point of a net

Let $(x_d)_{d\in D}$ be a net net of real numbers. Limit superior of a net is defined as $$\limsup x_d = \lim_{d\in D} \sup_{e\ge d} x_e = \inf_{d\in D} \sup_{e\ge d} x_e.$$ See, for example, Limsups ...
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3answers
35 views

Let $a_{n}$ be a sequence which converges to $c$. Then $c$ is a limit point of $a_{n}$ and it is its unique limit point.

Let $(a_{n})_{n=m}^{\infty}$ be a sequence which converges to a real number $c$. Then $c$ is a limit point of $(a_{n})_{n=m}^{\infty}$, and in fact it is the only limit point of $(a_{n})_{n=m}^{\infty}...
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0answers
12 views

Is the bound of different quotients the same as that of weak derivatives?

In $\S 5.8.2$ of Evan's PDE, there is a theorem relating to different quotients and weak derivatives. Theorem 3 (ii) Assume $1 < p < \infty$, $u \in L^p(V)$, and there exists a constant $C$ ...
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1answer
57 views

When is $\limsup_{n \to \infty} (a_n+b_n) = \limsup_{n \to \infty} a_n +\limsup_{n \to \infty} b_n$?

I know that $$\limsup\limits_{n \to \infty} (a_n+b_n) \le \limsup\limits_{n \to \infty} a_n +\limsup\limits_{n \to \infty} b_n.$$ But what should apply to A and B if we should have "=" ? I can't find ...
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0answers
9 views

Prove about sequence of sets and the characteristic function.

I'm trying to prove: If $(E_{n})_{n\geq 1}$ is a sqeuence of sets of $X$. We define $D_{1}=E_{1}$ and $D_{n}=D_{n-1}\bigtriangleup E_{n}$ for every $n \in \mathbb{N}$ then $\varliminf D_{n}=\...
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1answer
25 views

Show that $\limsup_{n\to\infty}\frac{x_{n+1}-x_1}n=\limsup_{n\to\infty}\frac{x_n}n$

Let $(x_n)_{n\in\mathbb N}\subseteq[-\infty,\infty)$. What's the easiest way to show that Show that $\limsup_{n\to\infty}\frac{x_{n+1}-x_1}n=\limsup_{n\to\infty}\frac{x_n}n$? Clearly, since $x_1/n\...
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0answers
20 views

Function defined on an interval $(a, \infty)$ having argument equal to $a$

In the definition for limit superior of real functions on this page we have: Let $f : (a, \infty) \to \mathbb{R}$. The limit superior as $x \to \infty$ is defined as $\displaystyle{\limsup_{x \to \...
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2answers
18 views

Limit Superior and Limit Inferior in Probability

Let $\{ X_{n} \}_{n=1}^{ \infty}$ denote sequence of random variables with the same distribution on $ \Omega $ and also: $$ A_{n} = \{ \omega \in \Omega : \ X_{n} ( \omega) \leq \varepsilon \log (n) \...
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1answer
16 views

$V_{\epsilon}(x^*)$ notation

What does the following notation $V_{\epsilon}(x^*)$ in the context and there are infinitely many terms of $a_n$ in $V_{\epsilon}(x^*)$ on this link.
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0answers
15 views

sup$_{x\in S}x$ notation for lim sup/inf

What does the following notation 'sup$_{x\in S}x$' mean in the context of limit superior and inferior? I am asking because usually I see it used like 'sup$X$' for some set $X$, but here it seems as if ...
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1answer
21 views

Separate tails of sequences in defining lim sup/inf

Why is on the following link a limit superior of a sequence $$a_n=(-1)^n/n$$ defined separately for $n$ odd and even? Namely for $n$ odd its $A_n=1/(n+1)$ and for even its $A_n=1/n$. What does it even ...
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1answer
27 views

Assume that $\{y_n\}$ converges to $l$. Prove that $\liminf{x_n + y_n} = \liminf{x_n} + l$

Let $\{x_n\}$ and $\{y_n\}$ be two bounded sequences. Assume that $\{y_n\}$ converges to $l$. Prove that $\liminf{(x_n + y_n)} = \liminf{x_n} + l$. In the exercise we were asked to show that $\liminf{...
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0answers
17 views

Is this a valid proof for the supremum subsequential limit $\geq$ lim sup?

Let the supremum subsequential limit be $L$. Assume the contrary, that $L< \lim \sup$. Write $L +2 \epsilon= \lim \sup$ for $\epsilon >0$. If there are only finitely many $i$ such that $a_i \...
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2answers
40 views

Borell-Cantelli lemma

If $\sum_{n} P(|X_{n}|>n)<\infty$, then prove that the $\limsup_{n}$ $|X_{n}|/n \leq 1$ a.s. My approach Let $E_{n}=|X_{n}>n|.$ $\sum_{n}P(E_{n})<\infty$ implies $P(E_{n} \text{ i.o})=...
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0answers
32 views

Find the $\limsup$ and $\liminf$ of $\{x_{n}\}$

Compute using their definition the $\limsup$ and $\liminf$ of $\{x_{n}\}$ with $x_n:= y_1 + y_2 + \dotsc + y_n$ $y_n$ := \begin{cases} 0 & \text{if } n =3p \\ 1 & \text{if } n = 3p+1\\ -1 &...
8
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1answer
467 views

How do prove this integral, defined on a countable set with infinite limit points, converges to zero?

Consider a continuous $f:A\to[0,1]$ where $A\subseteq[0,1]$. Edit: I did not use @Mathworker21's answer because I assumed it would not give the result I am looking for. Now I change my mind. Here is ...
2
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1answer
32 views

Show $\lim_{n\to \infty}\sum_{k=1}^{n} r_{k}\chi _{A_{k}} = f$

My Attemp: Let $(X,\mathcal{M})$ a measurable space and $𝑓:X→[0,\infty]$ measurable and , $(r_{n})_{n \in {\mathbb{N}}}$ is a sequence $(0,\infty)$ such that $r_{n} \to 0$ and $\sum_{n=1}^{\infty} ...
0
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1answer
29 views

Let $\{x_n\}_{n \in\mathbb{N}}$ be a bounded sequence of real numbers. Let us define $y_n = \sup\{x_k : k \geq n\}$ and $z_n = \inf{x_k : k \geq n}.$ [duplicate]

Let $\{x_n\}_{n \in\mathbb{N}}$ be a bounded sequence of real numbers. Let us define, for each ${n \in\mathbb{N}},$ $$y_n = \sup\{x_k : k \geq n\}$$ and $$z_n = \inf\{x_k : k \geq n\}.$$ Prove that ...

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