For questions concerning the definition and properties of limit superior and limit inferior.

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I need help about limsup and liminf ordering [on hold]

Let $f,g\in C^{0}\left(\left[a,b\right],\mathbb{R}_{+}\right)$. Assume that $f\left(t\right)\leq g\left(t\right),\forall t\in\left[a,b\right]$. Does this imply that $\limsup_{t\rightarrow ...
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Filter,dual ideal,definition of $\liminf_I \lambda$

We have this http://shelah.logic.at/files/506.pdf definition of $\lim \inf_I \bar{\lambda}$ in 1.1(3): for $I$ a filter on $\kappa$ let $I^+=2^\kappa \setminus I$.$$\lim \inf_I ...
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34 views

Prove a limsup and liminf inequality.

Let $(a_n)_{n\in\mathbb{N}}$ be a bounded sequence and let $B_n = \frac{1}{n} \sum_{i=1}^n a_i$ for each $n \in \mathbb{N}$. Prove that $\liminf a_n \le \liminf B_n \le \limsup B_n \le \limsup a_n$. ...
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24 views

References for limit superior and limit inferior of functions

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}\cup\{\pm \infty\}$. I would like to find books for the following notions and their properties: 1) $\limsup_{x\rightarrow x_0}f(x), \liminf_{x\rightarrow ...
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1answer
16 views

Limit inferior implication

I'm having trouble with the following statement. If $M=\liminf_{t\uparrow b}|x(t)|$ then $\exists$ a sequence $t_n\rightarrow b$ such that |$x(t_n)| \leq M+1$ I take it that with the sequence ...
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152 views

Computing a limit of sequence

After applying l'Hopital rule twice, one sees that $$ \lim_{n\to \infty} n a ~e^{-an} =0 \qquad \qquad (a\in [0,1]) . $$ I would like to ask if someone can prove it using different way? Bests.
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Is $\limsup_{z\to z_0}f(z)=\limsup_{k\to\infty}f(z_k)$?

Let $\Omega\in\Bbb C$ open, $f:\Omega\to\Bbb R$ a generic function. Let $(z_k)_k\subset\Omega$ s.t. $\lim_{k}z_k=:z_0\in\Omega$. The question is the following: is true that $$ \limsup_{z\to ...
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1answer
31 views

Proof of Fatou-Lebesgue Theorem

Good evening everyone, how can I prove the following inequality? Let $f,f_n\in L(X,\mu)$ on a measure space with $0\leq f_n(x)\leq f(x)$. If $f,f_n$ are $\mu$-almost everywhere in $X$, then is ...
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32 views

Pointwise convergence of $c$-concave envelopes as the sequence of cost functions converges pointwise

On Remark 1.12, page 33 of Villani's Topics in optimal transport we find the following problem. It should be easy but it's turning me mad. Let $c:\mathbb{R}^n\times\mathbb{R}^n\rightarrow ...
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Finding two sequences with a limsup value

Find two sequences ⟨ X n ⟩ and ⟨ Y n ⟩ such that limsup(Xn + Yn )=E=Euler's constant and limsup Xn + limsup Yn =Pi=π . I couldn't ...
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Liminf Brownian Motion question

For this assignment I'm working on, I was able to prove that: $$\limsup_{t \rightarrow \infty} \frac{B_t}{\sqrt t} = \infty$$ where $B_t$ is a Brownian Motion. I'd like to be able to prove: ...
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2answers
42 views

Isn't the statement of the Fatou's lemma somewhat problematic?

My lecture notes define $\int f := \int f^+ - \int f^-$ provided both $\int f^{\pm}$ are finite. And then the Fatou's lemma is stated in the following way: Let $f_n$ be a sequence of integrable ...
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55 views

If $\limsup_n\sqrt[n]{a_n}=\frac{1}{r}$, then $\limsup_n\sqrt[n]{(n+1)a_{n+1}}=?$

If $\limsup_n\sqrt[n]{|a_n|}=\frac{1}{r}$, then $\limsup_n\sqrt[n]{|(n+1)a_{n+1}|}=?$ Can I separate the product of the second limit as ...
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3answers
63 views

If $\lim_{n\to \infty}a_n = a\in \mathbb{R}$ . Prove that $\limsup_{n\to \infty}a_n x_n=a\limsup_{n\to \infty}x_n$ .

Note: $x_n$ is a sequence which is not necessarily convergent. The following was my attempt. Since $\lim_{n\to \infty}a_n=a$ then $\limsup_{n\to \infty}a_n=a$ . Also ...
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Multiples of some set has density

Nathanson gives a proof in Elementary Methods in Number Theory (Theorem 7.14) that, if a set $S$ of positive integers has $$ \sum_{s\in S}\frac1s<+\infty $$ then the set of positive multiples of ...
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2answers
64 views

Excerise 12.2 from Ross Elementary Analysis

I'm having a little bit of difficulty proving this question: Prove that $ \limsup_{n->\infty} |S_n| = 0 ~~$iff$ ~\lim_{n->\infty}S_n = 0. $ What I have so far: $ (\Leftarrow)$ $ $Suppose $ ...
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1answer
26 views

Every sequence of sets have coverging subsequence

Is that true, that every sequence of sets have coverging subsequence? We say that sequence of sets $A_1, A_2, A_3, ...$ coverging iff ${\limsup}_{n \to \infty} A_n = {\liminf}_{n \to \infty} A_n$
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What is the limsup of the following sequence of sets?

Problem: Let $A_n=\{\frac{m}{n}:m\in\mathbb{Z}\}$. Find $\limsup_{n\rightarrow\infty}A_n$ and $\liminf_{n\rightarrow\infty}A_n$ Attempted: It is clear that limsup should be $\mathbb{Q}$. I can show ...
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1answer
46 views

Give an example to show that the inequalities are strict inequalities

Give an example to show that the following three inequalities $$\liminf_{n \to \infty} (a_n) +\liminf_{n \to \infty} (b_n)\le\liminf_{n \to \infty} (a_n+b_n)\le\limsup_{n \to \infty} (a_n+b_n) \le ...
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1answer
34 views

Doubt on Tail events and Kolmogorov Zero-One Law

In this wiki article on the law of the iterated logarithm, one states that, given $M>0$, the event $$A=\left\{\limsup_{n \to \infty} \frac{S_n}{\sqrt{n}} > M\right\}$$ has probability $0$ or ...
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51 views

how to show $\lim \sup_{n \to \infty} \frac{\phi(n+1)}{\phi(n)}= \infty$ and $\lim \inf_{n \to \infty} \frac{\phi(n+1)}{\phi(n)}= 0$?

Let $\phi (n)$ be the Euler's totient function . Thenhow to prove that the set $\{\dfrac{\phi(n+1)}{\phi(n)}: n \in \mathbb Z^+\}$ is unbounded above that is how to show $\lim \sup_{n \to \infty} ...
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1answer
32 views

Prove that $\limsup_{n \to \infty} (a_n+b_n)= a + \limsup_{n \to \infty} b_n$

I have to prove the following: Given that lim $a_n$ exists and that lim $a_n$ = $a\in \mathbb{R}$ prove that: $$\limsup_{n \to \infty} (a_n+b_n)= a + \limsup_{n \to \infty} b_n$$ I proved one ...
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1answer
21 views

limit superior and inferior of measurable sets

I am trying to prove the following: Let $(A_n)_{n \geq 1}$ be a sequence of measurable sets, then (i) $|\lim \inf_{n \to \infty} A_n| \leq \lim \inf_{n \to \infty} |A_n|$ (ii) If there is $n \in ...
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2answers
71 views

how that if $P(\lim \sup A_n) = 1$ then, $P(\bigcup_{n=1}^\infty A_n)=1$

Question: Let $\{A_n\}$ be a sequence of independent events in a probability space $(\Omega, F, P)$ show that if $P(\lim \sup A_n) = 1$ then, $P(\bigcup_{n=1}^\infty A_n)=1$ I tried solving this ...
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1answer
25 views

Proving a subsequence from lim inf

I'm trying to solve this problem, Let $a_n$ be a sequence such that lim inf$ |a_n| = 0 $. Prove that there is a subsequence $a_{n_k}$ such that $\sum a_{n_k}$ converges. So far, I tried to say we ...
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1answer
26 views

Prove absolute convergence for a summation

I need help with this problem. I've been staring at the page blankly tyring to think of ways to solve it. Any hints/solutions would be greatly appreciated. If $ \displaystyle \lim_{n \to ...
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1answer
42 views

Prove equality in liminf of set

I need help in proving $\chi_{\liminf A_n} = \liminf \chi_{A_n}$ Where $\liminf A_n=\bigcup \limits_{n=1}^{\infty}\bigcap \limits_{k=n}^{\infty}A_k$ and $\chi_{A}$ is the characteristic ...
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A quick question on the limsup of sets and logical connectives

I know we can interpret $x\in\limsup{A_n}$ iff $x$ is in infinitely many of the $A_n$'s, but am confused as to which logical connective we need to use to describe when $x\in\limsup{A_n}$, when we ...
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2answers
40 views

liminf inequality in measure spaces

Let $(X;\mathscr{M},\mu)$ be a measure space and $\{E_j\}_{j=1}^\infty\subset \mathscr{M}$. Show that $$\mu(\liminf E_j)\leq \liminf \mu(E_j)$$ and, if $\mu\left(\bigcup_{j=1}^\infty ...
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1answer
48 views

Find $\lim \sup A_n$ and $\lim \inf A_n$?

Question: Let $\Omega = R^2. A_n$ is the interior of a circle with center at $\{\frac{(-1)^n}{n},0 \} $ at radius 1. Find $\lim \sup A_n$ and $\lim \inf A_n$? My answer is the following; Let $\Omega ...
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37 views

Proving that $\mathcal{L}(x_n+y_n) \subset {\cal L}(x_n)+{\cal L}(y_n)$.

Let $(x_n)$ and $(y_n)$ be two bounded real sequences. Let: $${\cal L}(x_n) =\{ L \in [-\infty,+\infty] \mid L \text{ is the limit of some subsequence }(x_{n_k}) \},$$ and the same for ${\cal L}(y_n)$ ...
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1answer
21 views

Need help with: $\lim \sup x_n = \sup{z_k}$ where $z_k$ is a set of limit points

This question stems from another one, but presents my concern in a more specific way. There is a theorem saying that: $\lim \sup x_n = \sup{z_k}$ where $z_k$ is a set of limit points for the ...
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1answer
19 views

$\lim \sup x_n = \sup{z_k}$ where ${z_k}$ is a set of limit points for the sequence ${x_n}$

There is a theorem saying that: $\lim \sup x_n = \sup{z_k}$ where ${z_k}$ is a set of limit points for the sequence ${x_n}$. All sets and sequences are real. Limit point for a sequence is a point ...
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1answer
29 views

Weakest assumption for $\lim \sup x_n \in \mathbb R$

What is the weakest assumption to be satisfied so that $\lim \sup x_n \in \mathbb R$. The same question for $\lim \inf x_n$. Note that $\mathbb R$ does not include $\pm \infty$. Should $x_n$ ...
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If expectation is bounded away from 0, is then the probabilty of being positive also positive?

Consider a discrete-time stochastic process $(Y_t)_{t \geq 0}$, where we know that $$\liminf_{t \rightarrow \infty} \mathbb{E}[Y_t] >0.$$ Does this imply that $$\liminf_{t \rightarrow \infty} ...
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1answer
41 views

$\liminf A_n$ and $\limsup B_n$

Show that for a sequences of sets $A_n$ and $B_n$ $\liminf A_n\cap\limsup B_n \subset\limsup(A_n\cap B_n)$ Can you give some hint please.How can ı show this question. Thank you
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38 views

Question about $\liminf$ and $\limsup$

I have this part: Where $$\underline{F(+\infty)}=\liminf_{x\rightarrow +\infty} F(x), \overline{F(-\infty)}=\limsup_{x\rightarrow-\infty} F(x)$$ My question is: How does property $(3.12)$ follow ...
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1answer
25 views

Let $(s_n)$ and $(t_n)$ be bounded sequences of nonnegative numbers. Prove $\lim \sup{s_n,t_n}$ $\leq$ $(\lim \sup{s_n})(\lim \sup{t_n})$?

Let $(s_n)$ and $(t_n)$ be bounded sequences of nonnegative numbers. Prove $\lim \sup{s_n,t_n}$ $\leq$ $(\lim \sup{s_n})(\lim \sup{t_n})$? Can someone help explain this proof to me?
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58 views

Is $\limsup f(x) \le \liminf g(x)$ if $f(x) \le g(x)$ for all $x$

I was wondering if the following is true. $\limsup_{x \to \infty} f(x) \le \liminf_{x \to \infty} g(x)$ if $f(x) \le g(x)$ for all $x$. One more constrain we can impose is that $g(x)$ and $f(x)$ are ...
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55 views

Show that $\limsup|Y_{1}+…+Y_{n}|/n = \infty$ almost surely

Can someone help me with part c) of question 2.8 located here (a 2005 probability course from Warwick University): https://homepages.warwick.ac.uk/~masgav/teaching/pm05_sheet2.pdf The question is: ...
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1answer
40 views

Simplify the function of x

I am given $$ f(x) = \lim\limits_{n \to \infty } (1+x)(1+x^2)(1+x^4)...((1+x^{2^n})$$ where $|x| <1$ I think maybe we may apply squeeze theorem here to simplify expression so I took log of the ...
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1answer
43 views

Telescoping Sum of expectations: limsup exists but limes not necessarily

Let $X_t$ for $t \in \{0, 1, \dotsc, \}$ be a sequence of non-negative integer-valued random variables. Suppose that $$\mathbb{E}[X_t - X_{t+1} \mid X_t>0 ] \leq c \quad \text{ for some constant ...
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1answer
30 views

Telescoping Series: If only $\liminf$ does exist, and not $\lim$

Consider the following telescoping series: $$S:=\sum_{t=0}^{\infty} (x_t - x_{t+1}).$$ If $$\lim_{t \rightarrow \infty} x_t=0,$$ then this simplifies to $$S=x_0 - \lim_{t \rightarrow \infty} ...
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1answer
38 views

show that $ A_n \cup B_n \to A \cup B$ and $ A_n \cap B_n \to A \cap B$

Question: if $ A_n \to A $ and $ B_n \to B $, show that $ A_n \cup B_n \to A \cup B$ and $ A_n \cap B_n \to A \cap B$ My solution way is the following; $$ \lim_{n\to \infty} A_n = A $$ and ...
2
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1answer
61 views

limit supremum and infimum question

Question: Show that $\limsup A_n -\liminf A_n = \limsup(A_n A^c_{n+1}) =\limsup (A^c_n A_{n+1})$ the thing I understand from this queston is the following; $$\bigcap_{n=1}^\infty ...
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1answer
13 views

Let $L, M$ be real numbers, and let $s_n$ and $t_n$ be sequences such that $L \leq s_n \leq M$ and $L \leq t_n \leq M$ for all $n$ Define:

I understand Part (a) just fine, but I am lost as to how to prove the two inequalities given. Can anyone help me out here?
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3answers
83 views

Ratio test with limsup vs lim

Could I prove that the ratio test still works using $\limsup(\frac{a_{n+1}}{a_n})$ instead of $\lim(\frac{a_{n+1}}{a_n})$? I think for $\limsup<1$ I could show that for $\epsilon>0, N>1 ...
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1answer
36 views

Series limsup converges

This showed up as an optional challenge problem from my class: Show that $\displaystyle \sum_{n=1}^\infty a_n$, $a_n>0$ converges if $\limsup(\sqrt[n]{a_n})<1$ and diverges if ...
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3answers
68 views

A sequence in which liminf < limsup

In this question we are looking for a sequence in which liminf is strictly less than limsup. A majority of the examples that I came up with and found were in which liminf<=limsup. As for the second ...
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0answers
22 views

passing to a liminf from weak convergence. [duplicate]

Say i have a sequence $(||x_n||)$ which is bounded above, and say $x_n$ converges weakly to $x$. Then, how can I show that $||x|| \le \liminf_{n \to \infty}||x_n||$. Well, clearly, the $\liminf$ ...