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

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4
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1answer
50 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 ...
1
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3answers
61 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 ...
3
votes
0answers
33 views

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 ...
1
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2answers
58 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 $ ...
1
vote
1answer
24 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 ${\limsup}_{n \to \infty} A_n = {\liminf}_{n \to \infty} A_n$
1
vote
1answer
30 views

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 ...
1
vote
1answer
43 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 ...
0
votes
1answer
30 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 ...
2
votes
0answers
48 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} ...
1
vote
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 ...
0
votes
1answer
19 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 ...
2
votes
2answers
67 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 ...
1
vote
1answer
24 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 ...
1
vote
1answer
24 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 ...
2
votes
1answer
34 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 function of ...
0
votes
0answers
23 views

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 ...
0
votes
2answers
34 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 ...
0
votes
1answer
46 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 ...
3
votes
2answers
36 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 ${\cal L}(y_n)$ ...
0
votes
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 ...
1
vote
1answer
17 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 ...
-1
votes
1answer
28 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 $x_n$ ...
0
votes
0answers
13 views

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} ...
2
votes
1answer
39 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
0
votes
1answer
37 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 ...
0
votes
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?
0
votes
2answers
57 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 ...
1
vote
2answers
52 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: ...
0
votes
1answer
39 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 ...
1
vote
1answer
36 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 ...
2
votes
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} ...
0
votes
1answer
37 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 ...
0
votes
1answer
12 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?
0
votes
3answers
70 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 ...
0
votes
1answer
35 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 ...
2
votes
3answers
62 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 ...
0
votes
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 $\liminf$ ...
1
vote
1answer
39 views

Showing that $\left\{ {\mathop {\lim }\limits_{n \to \infty } {X_n} = X} \right\}$ is an event

Let $X$ and ${\left( {{X_n}} \right)_{n \in \mathbb{N}}}$ be random variables on a measurable space $\left( {\Omega ,\mathcal{F}} \right)$. Show that: 1) $\left\{ {\omega \in \Omega :\mathop {\lim ...
1
vote
1answer
46 views

lim sup (liminf) (An\Ak)

I need to show that $\limsup_n \liminf_k A_n \cap A_k^c=\phi$. Thus $\bigcap_n\bigcup_{r\geq n} \bigcup_k \bigcap_{m\geq k} A_r\cap A_m^c=\phi$? I am trying to show that $\lim_n P(\liminf_k A_n ...
6
votes
1answer
144 views

Prove $\limsup_{n\to\infty}|\{(p,q)\in T\times T,p!q!=n\}|=6$

Let $T$ be the set of nonnegative integers, I need to prove that $$\limsup_{n\to\infty}|\{(p,q)\in T\times T,p!q!=n\}|=6$$ It's really easy to show that $$\limsup_{n\to\infty}|\{(p,q)\in T\times ...
1
vote
1answer
41 views

Question on lim sup

Given that $(a_n)$ is a bounded sequence. If $b_n=\sup\{a_n,a_{n+1},a_{n+2},a_{n+3},\ldots\}$ prove that $\lim\sup(a_n)$ is a limit of $b_n$. The lim sup is defined as l.u.b. of $$A = \{x\in R| \ x ...
1
vote
1answer
60 views

Set lim inf and lim sup question.

If $A_n = \{m/n : m \in N\}, n \in N$. What is lim $\inf_n$ $A_n$ and lim $\sup_n$ $A_n$? Using the definitions: lim inf $A_n$ = $\bigcup_{n\in N}$ $\bigcap_{k > n}$ $m/n$ = $\cup_{n\in N}$ $0$ = ...
1
vote
1answer
29 views

$\lim \sup, \lim \inf $ of a fraction of sequence

Let $x_n$ be an upper bounded sequence, $y_n$ is a sequence that $\forall n \in \mathbb{N}$ such that $y_n \rightarrow y>0 $ then prove that: $$ \lim \sup \frac{x_n}{y_n} = \frac{\lim \sup ...
1
vote
1answer
62 views

$\liminf$ of a sequence of functions

I know that, given $f(x)$, $\inf f(x)$ is its greatest lower bound. I also know that $\lim \inf f(x)$ is its greatest lower bound when $x \longrightarrow \infty$. For a sequence of functions $f_{n}$, ...
1
vote
1answer
64 views

Properties of sup and lim inf.

Let $(a_{n})_{n \geq1}$ be a sequence of numbers such that $a_n\leq M$ for all $n \geq 1$ . Prove that $$ \lim_{n\to\infty} \inf \{a_n,a_{n+1},...\} = \sup_{n \geq1} \inf\{a_n,a_{n+1},...\} $$ ...
0
votes
0answers
54 views

Finding limsup and liminf for odd and even $A_n$

I am trying to understand $\limsup$ and $\liminf$. I have this homework problem: For each natural number $n$, let $A_n=[0,1]$ if $n$ is odd, and $A_n=[1,2]$ if $n$ is even. Find both ...
0
votes
0answers
13 views

Proof verification (limit inferior) is needed.

Could please somebody verify the following proof? We have a sequence of real numbers $x_n$ such that $x_n \ge f(t- \epsilon)-p(\epsilon,n)$, $p(\epsilon,n)$ $\forall \epsilon$ and $\forall n$, ...
1
vote
3answers
33 views

lim sup iff conditions - please help explain

Deciphering the definition of the upper limit, we see that $\limsup x_n=L$ is and only if the following two conditions are fulfilled: (a) $\forall\epsilon>0\;\;\exists N\in\mathbb{N}$ such that ...
2
votes
1answer
67 views

Confused about this definition of limit superior.

The definition I am given is as follows: Let $(x_n)$ be a real valued sequence. For each positive integer $n$, let $s_n:=\sup\{x_m:m\geq n\}$. If $(s_n)$ converges, we denote its limit by ...
2
votes
0answers
13 views

Limsup question $ \sum_{n=1}^N \sum_{m=1}^N x_{m-n} \leq N^2 \bigg( \limsup_{n \to \infty} x_n + o(1) \bigg) $

Reading notes on the Poincare Recurrence Theorem and I am a bit stuck with Theorem 1. For a measure preserving dynamical system $T: X \to X$ we have $\mu(T^{-1}E) = \mu(E)$. Why does it "easily ...