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

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Prove that lim sup of a function belongs to a certain sigma algebra

I am so baffled with this problem: Let $B$ be a standard Brownian motion, $\{ \mathcal{F}_t \}$ be the filtration generated by the Brownian motion. I would like to show that for any $k>0$, ...
2
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2answers
47 views

The limit of an expected value vs expected value of a limit in this betting game

Setting The outcome $X$ of a slot machine takes values 1,2,or 3 with probability $p(1) = \frac{1}{2}$, $p(2) = \frac{1}{4}$, $p(3) = \frac{1}{4}$. We are given 3 for one odds, that is if we bet 1 ...
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2answers
42 views

lim sup, lim inf, and inequalities

Suppose we have two sequences ${a_n}$ and ${b_n}$, which satisfies $ a_n \le b_n$ for $n=1,2,3,\ldots$. Do we have the following inequalities to be true? $$\limsup_{n \to \infty} a_n \le \limsup_{n ...
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1answer
39 views

Limsup and liminf of a function

Let $k\in(0,1)$ be fixed and $L\in \mathbb{R}$ is finite. If $\limsup_{x\to\infty}f(kx)=L$ and $\liminf_{x\to\infty}f(\frac{x}{k})=L$ then is it possible to say $$\lim_{x\to\infty}f(x)=L.$$
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1answer
70 views

If $a_n \to a < 0$, then $\lim \inf a_nb_n = a \lim \sup b_n$

I got this proof that I can't show for some hours now. Does anyone have a hint? If $a_n, b_n$ are bounded sequences and $\lim_{n\to\infty} a_n = a < 0$ then $$\liminf_{n\to\infty} a_nb_n = a \cdot ...
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2answers
34 views

confusing definition of lim sup

My textbook defines: \begin{equation} \limsup (a_n) = \min\{M ∈ R |\ \ ∃n_0 \ \ ∀n > n_0, a_n ≤ M\}. \end{equation} And it gives an example: Let $a_n$= 1+ $\frac{1}{n}$. Then $\limsup (a_n)$ = 1. ...
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2answers
30 views

limsup of fraction ratio to limsup of the square of ratio

Let $({a_n})_{n\geq 0}$ be non decreasing in $n$. How to show that $$\limsup \frac{a_n}{n} = \limsup \frac{a_{n^2}}{n^2} ?$$
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1answer
97 views

Limit based on bounds

Let $k\in(0,1)$ be fixed. If $\limsup_{x\to\infty}f(kx)\leq \epsilon$ and $\liminf_{x\to\infty}f(\frac{x}{k})\geq -\epsilon$ for all $\epsilon>0$ then is it possible to say ...
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1answer
23 views

limsup of average smaller than limsup

I have read this solution, but I could not understand it. It has shown $$\sigma_n\leqslant \frac 1n\sum_{j=1}^ks_j+\sup_{l\geqslant k}s_l,$$ but how does it go to $$\sup(\sigma_n)\leqslant \frac ...
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2answers
26 views

Show every bounded infinite set has a maximum limit point and a minimum limit point.

Show every bounded infinite set has a maximum limit point and a minimum limit point. Here is my thought even if it is not formal Let $S$ be bounded and infinite set. Bolzano–Weierstrass theorem: ...
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1answer
51 views

Prove $\liminf a_n + \liminf b_n\le \liminf (a_n +b_n) $ [duplicate]

$a_n$ and $b_n$ are two bounden sequences Prove $$\liminf(a_n) + \liminf(b_n)\le \liminf(a_n +b_n)$$ Should I use $$\inf(a+b) = \inf(a) + \inf(b)$$ and i could not come up with how to proceed from ...
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1answer
27 views

For $s_n$ a sequence in $\Bbb R$, if $\lim s_n$ defined as a real number, then $\liminf s_n = \lim s_n = \limsup s_n$.

I'm studying some real analysis, and I'm trying to figure out how to prove the following theorem. For the most part, I've got things figured out. I'll post the theorem and my proof (which closely ...
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1answer
49 views

Finding limits superior and inferior of a sequence

Find the limits superior and inferior of the following sequences: note: "For a set, those are the infimum and supremum of the set's limit points, respectively. " $a_n=\frac{n}{n+1} ...
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2answers
38 views

properties of partial sums of series $\sum \sin(n)$

There have been many questions posted here about the sequence $s(x)_n = \sin(nx)$ and its corresponding series $S(x)$. In particular, it has been shown that $s(x)_n$ only converges if $x$ is a ...
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1answer
28 views

Prove that if lim sup$_{n} n^{2}a_{n} = 0 \Rightarrow \sum_{k = 1}^{\infty} a_{k}$ converges

Prove that if lim sup$_{n} n^{2}a_{n} = 0 \Rightarrow \sum_{k = 1}^{\infty} a_{k}$ converges. Also assume that the sequence is positive. lim sup$_{n} n^{2}a_{n} = 0$ means that for every $\epsilon$, ...
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1answer
43 views

How can I prove that a sequence such that every converging subsequence coverges to the same limit, converges?

I want to claim that if $(x_n)_{n\in N}$ is a sequence, and there is $a$ such that if $(x_{n_k})$ converges, so $\lim x_{n_k} = a$ (it means that all converging subsequences have the same limit), then ...
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1answer
14 views

Operator which is invariant in terms of left shift

Let $l:l^{\infty}(\mathbb R)\rightarrow \mathbb R$ be a linear map( where $l^{\infty}$ is the set of bounded sequences) such that the following holds: (i) $l(Tx)=l(x)$, where T is the left shift ...
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3answers
59 views

A question corcerning limsups and liminfs of sets

Let $\{E_n \}$ be sets indexed by natural numbers. MY books define the following notions: $$ \limsup E_n = \bigcap_{k=1}^{\infty} \bigcup_{n=k}^{\infty} E_n$$ $$ \liminf E_n = ...
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1answer
29 views

prove $\limsup_{n\to\infty}\frac{\sqrt{2({{x_1}^2+{x_2}^2+…+{x_n}^2})}}{n}\leq\limsup_{n\to\infty}\frac{x_n}{\sqrt{n}}$

Let $x_1,x_2,x_3$,... be a sequence of nonnegative real numbers. Prove that $\limsup_{n\to\infty}\frac{\sqrt{2({{x_1}^2+{x_2}^2+...+{x_n}^2})}}{n}\leq\limsup_{n\to\infty}\frac{x_n}{\sqrt{n}}$ I ...
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1answer
42 views

lim sup of set, elementary set theory

I am referring to the post: lim sup and lim inf of sequence of sets. how would we prove that for $\limsup A_n $ $x$ is in infinitely many sets $A_i$? If I define set $A$ as a set that consists of ...
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0answers
13 views

find subsequential, lim sup and lim inf

find the subsequential of {[1+(-1)^n]n + (1/n)} by using the limit theorem, I have: {2n + 1/n } as n -> infinity and n even then is 2 {0*n + 1/n} as n -> infinity and n odd then is 0 so the ...
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0answers
31 views

Monotone Sequence of Sets

If someone can check my proof for the following statement, would be awesome. Thanks. Suppose $\{A_n\}$ is a monotone sequence of subsets. If $A_n \downarrow$, then $\lim_{n \rightarrow \infty} A_n = ...
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0answers
49 views

limit superior and limit inferior sequence

Find limit superior and limit inferior of $\frac{n+(-1)^n n^2}{n^2+1}$: the subsequential limit is $1$ and $-1$. so the limit $\sup = 1$ and limit $\inf = -1$ let $E_k = \{a_n\mid n>k\}$. $$E_k= ...
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0answers
36 views

Proofs Regarding Open and Closed Sets

I need to prove the following regarding open and closed sets: 1. A set L is closed iff for any converging sequence $(x_n)$ with $x_n\in L$, the limit $x=\lim_{n\to\infty}{x_n}$ is also an element of L ...
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1answer
31 views

Subsequence converging to inf of sup

Let $(x_n)$ be a bounded sequence, and for each $n\in\mathbb{N}$ let $s_n=\sup\{x_k:k\geq n\}$ and $S=\inf\{s_n\}$. I need to show that there exists a subsequence of $(x_n)$ that converges to $S$. I ...
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32 views

Limsup and liminf comparisons

I'm studying for my analysis midterm and ran across the following in my notes: For a sequence $\{x_n\}$ in $\mathbb{R}$, define $b_k = \sup \{x_{k+1}, x_{k+2}, \ldots\}$ and $a_k = \inf \{x_{k+1}, ...
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0answers
31 views

Help understand part of the proof. Radius of convergence is $\frac{1}{\limsup |a_n|^{1/n}}$

Can you help me understand the highlighted parts of the proof. Thanks :) Theorem: Let $\sum{a_nz^n}$ be a power series, let r be its radius of convergence. Then $\frac{1}{r} = \limsup |a_n|^{1/n}$. ...
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2answers
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there are only finitely many n with $v+ϵ<x_n$ and $2)$ there are infinitely many $n$ with $v−ϵ<x_n$

Prove that if $v$ is the limit superior of a bounded sequence $X$, then for any $\epsilon>0,$ $(i)$ there are only finitely many n with $v+ϵ<x_n$ and $2)$ there are infinitely many $n$ with ...
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1answer
22 views

Showing the equivalence of different definitions of the box-counting dimension

I am trying to prove the statment "by taking logarithms...". $\lim\limits_{\delta \rightarrow 0} \frac{log(N_{4\delta}(F))}{log(\frac{1}{4 \delta})} \leq \lim\limits_{\delta \rightarrow 0} ...
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74 views

Limit Superior and Limit Inferior of sequence

I am taking an introductory course in Real Analysis and $(i)$ My textbook gives the following definition of a limit superior $U$ which satisfies the following conditions: $(a)$ For every ...
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3answers
86 views

Is it true that $\lim \inf (x_n - y_n ) = \lim \inf (x_n ) + \lim \inf ( - y_n )$ if $(x_n)$ is convergent?

Consider the sequences $x_n$,$y_n\in\mathbb R$. We know that $x_n$ converges, but absolutely nothing is mentioned about $y_n$. I saw in a proof the following claim and I don't get the reasoning : $$ ...
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0answers
28 views

Is lim sup $s_{n}$ larger than all other tails of $s_{n}$?

I was reading a proof and it said that let $q$ = lim sup $s_{n}$. Then q is the largest possible value any $s_{n}$ in a tail of the sequence of $s_{n}$ can attain. So, sup $T_{m} \leq q$ where ...
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36 views

A General Question arises about norms and limits [closed]

Suppose $x_n \to x_0$, i.e. $\lim x_n=x_0$. How we can show that $||x_n|| \to ||x_0||$, that is $\lim_{n \to \infty} ||x_n||=||x_0||$, so $\lim_{n \to \infty} ||x_n|| = ||\lim_{x \to \infty} x_n||$?
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1answer
29 views

limsup of sequence of random variables scaled with n^{-1}

I have been stuck on the following problem: Let $X_i$ iid non-negative random variables (not only necessarily integer-valued) and define $ A:=\limsup \frac{X_i}{i}$. Prove that ...
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1answer
47 views

show there is a subsequence that converges to $\lim_{x_k \rightarrow \infty} \inf x_k $

For a sequence $(x_k) \in \mathbb{R}$ , $\lim_{k \rightarrow \infty} \sup x_k = \lim_{k \rightarrow \infty}$ $(\sup{x_l| l ≥ k})$ $\lim_{k \rightarrow \infty} \inf x_k$ = $\lim_{k \rightarrow ...
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Inequality involving a strictly positive sequence

I was asked to prove the following : \begin{array}{l} x_n > 0,\,\forall n \in Z^ + \\ \lim \sup (\frac{{x_{n + 1} + x_1 }}{{x_n }})^n \ge e \\ \end{array} Is my approach correct ? Using ...
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1answer
31 views

Prove that the square root and exponent of a function in a $\limsup$ equals the the square root and exponent of a $\limsup$ of the function?

By what property do the following equalities hold? \begin{align*} \frac{1}{\limsup\limits_{n \to \infty} \sqrt[2n]{\left|a_n\right|}} = \left( \frac{1}{\limsup\limits_{n \to \infty} ...
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1answer
35 views

Limes superior of an ergodic and stationary process

Given a stationary and ergodic sequence of real-valued random variables $(X_n)_{n \geq 0}$ and a Borel$(\bar{\mathbb{R}})$-set A, with $\mathbb{P}[X_0 \in A] > 0$, I want to show, that ...
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1answer
32 views

Corollary of the Kolmogorov Zero-One Law, proof

Here is an application of the Kolmogorov Zero-One Law given in my textbook (a probability path by Resnick page 107-108). It states that the random variables $\limsup_nX_n$ and $\liminf_nX_n$ are ...
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2answers
74 views

If $\limsup x_n = x$, $\lim y_n = y$, $x_n, y_n > 0$, then does $\limsup (x_n y_n)= xy$? [duplicate]

I have to prove the following statement, but I can't. If $\limsup x_{ n }=\, x,\lim y_{ n }=\, y, \, x_{ n },y_{ n }>0$, then $\limsup (x_{n}y_{n})=xy$. Will you give me some hint or solution?
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1answer
53 views

Prove that $\limsup\{x_{n}+y_{n}\} =\lim_{n\rightarrow\infty}x_{n}+\limsup\{y_{n}\}$ [closed]

Assume $\lim_{n\rightarrow\infty}x_{n}$ exists. Prove that for any sequence $y_n$, we have $$\limsup\{x_{n}+y_{n}\} =\lim_{n\rightarrow\infty}x_{n}+\limsup\{y_{n}\}$$ I got stuck on this question ...
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0answers
36 views

tricky question regarding Series, Limits and Convergence

Suppose we have 2 real power series for every $k=1,2,3,\ldots$ $$ f_k(x)=\sum_{n\in\mathcal{S}^{(+)}(k)} x^n d_n(k),\quad g_k(x)=\sum_{n\in\mathcal{S}^{(-)}(k)} x^n d_n(k), \quad d_n(k)\geq 0 $$ where ...
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0answers
44 views

Limit superior does not increase when a sequence is replaced by its sequence of averages [duplicate]

Let $\{x_n\}$ be a bounded sequence of real numbers, and define a new sequence $\{\sigma_n\}$ by $$\sigma_n=\frac 1n\sum_{i=1}^nx_i.$$ Prove that $\limsup \sigma_n\le \limsup x_n$. I am confused on ...
1
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1answer
82 views

Show that $ \limsup x_n ≤ \limsup y_n$ and $\liminf x_n ≤ \liminf y_n$

Let $(x_n)$ and $(y_n)$ be bounded sequences such that $x_n ≤ y_n$ for all $n \in \mathbb{N}$. Show that $\limsup x_n ≤ \limsup y_n$ and $\liminf x_n ≤ \liminf y_n$.
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1answer
44 views

limsup of the product of two sequences, of which one converges

Let $\{a_n\}_{n=1 }^{\infty}$ and $\{b_n\}_{n=1}^{\infty} $ be two sequences in $\mathbb{R}$, with the first sequence convergent . Prove that $$ \limsup\limits_{n\to \infty} a_n b_n ...
1
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1answer
46 views

Is there a contradiction in these two definitions of limit Superior?

Definition 1 : Let $A=\{a_n\}$ be a sequence of real numbers not necessarily bounded . Then we define : $\lim \sup ~ a_n = \inf ~\{\sup ~a_n,~\sup ~a_{n+1}~\sup ~a_{n+2}, \cdots\} $ Definition 2 : ...
5
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2answers
94 views

How can we apply the Borel-Cantelli lemma here?

Let $(A_n)$ be a sequence of independent events with $\mathbb P(A_n)<1$ and $\mathbb P(\cup_{n=1}^\infty A_n)=1$. Show that $\mathbb P(\limsup A_n)=1$. It looks like the problem is practically ...
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1answer
91 views

How to find limsup and liminf for sequence of sets

Given $A_n$ $=$ {$w$$|$$0$ $\le$$w$$\le$$1$$-$$\frac{1}{n}$} Find $\limsup_{n \to \infty}$$A_n$ $\text{and}$ $\liminf_{n \to \infty}$$A_n$. Can anyone guide me on how to solve this question? I ...
2
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0answers
106 views

Grasping Lim Sup and Lim Inf

The intuitive picture Even tho i had proven most things about limit superior and limit inferior, i was struggling in getting an intuitive and big-picture of limit superior and limit inferior of a ...
1
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1answer
47 views

Finding the limsup and liminf of a sequence of disks

Let $A_n$ be the interior of the circle with center at $( (-1)^n/n,0) )$ and radius $1$. In other words, $A_n$ = { $ (x,y) | (x -(-1)^n/n )^2 + (y -0)^2 < 1$}. What is the $\limsup_n A_n$ and ...