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

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Exercise from Williams book Probability with martingales

I'm doing this exercise from Williams book Probability with martingales Let $(X_n)$ be a sequence of IID random variables with $E(|X_n|) = \infty $ for all $n$. Then prove that $$1)\ \sum_n ...
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27 views

Prove: $\limsup_{n\to\infty} (a_n)\leq \sup(a_n)_{n=1}^\infty$

Proposition: Let $(a_n)_{n=1}^\infty$ be a sequence of real numbers. Show that $\limsup_{n\to\infty} (a_n)\leq \sup(a_n)_{n=1}^\infty$. Any hints on how to prove it will be appreciated.
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meaning of limsup

Let $(a_n)_{n=1}^\infty$ be a sequence of real numbers. Let $$L^+=\lim \sup(a_n)_{n=1}^{\infty}$$ Is it true that there exists an N such that $a_n\leq L^+,\ \ \forall n\geq N$? I was trying to ...
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41 views

Computing lim sup and lim inf of $\exp(n\sin(\frac{n\pi}{2}))+\exp(\frac{1}{n}\cos(\frac{n\pi}{2}))$ and $\cosh(n\sin(\frac{n²+1}{n}\frac{\pi}{2}))$?

It's the first time I encounter lim sup and lim inf and I only just know about their definitions. I have difficulties finding out about lim sup and lim inf of the following sequences ...
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17 views

Is there a Heine cretierion of liminf of a function?

Lately i've been struggling with understanding the meaning of $\liminf_{x\to x_0}f(x)$ assuming $f:X\to\mathbb C$ for $X$ a metric space, or for that matter $f:\mathbb R\to \mathbb R$. Could you give ...
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2answers
61 views

Is $\lim\sup=\sup\lim$?

Assume $(a_n(x))_{n=1}^{\infty}$ is a bounded sequence in $\mathbb R$, when $x$ is $\in\mathbb R$ and is relevant to the sequence in some way that doesn't really interest us in my question. Assume ...
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39 views

the relation between the lower limit of set and the lower limit of function

Define the characteristic function of set $A$ to be $${\chi _A}(x) = \left\{ {\begin{array}{*{20}{c}} 1&{x \in A}\\ 0&{x \notin A} \end{array}} \right..$$ For any given collection of sets ...
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1answer
70 views

Showing that lim sup of sum of iid binary variables $X_i$ with $P[X_i = 1] = P[X_i = -1] = 1/2$ is a.s. infinite

Let $(X_i)_{i\in\mathbb{N}}$ be an i.i.d. sequence of binary random variables with $$P[X_i = 1]=P[X_i = -1] = \frac{1}{2}$$ and let $$S_n = \sum_{i=1}^{n} X_i.$$ I'd like to show that $$P[\lim ...
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1answer
52 views

Proving that $\liminf_{n\to\infty} x_n = \sup\left\{z : \{n : x_n < z\} \text{ is finite}\right\}$; what does the answer mean?

Suppose $x_n$ is a bounded sequence. Prove that $$\liminf_{n\to\infty} x_n = \sup\left\{z : \{n : x_n < z\} \text{ is finite}\right\}$$ I'm having trouble even interpreting what this is suppose to ...
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1answer
63 views

Violation of Fatou's theorem?

I have an assignment where the answer I have come up with violates Fatou's lemma. Clearly, I have either reached the wrong answer or misunderstood Fatou's lemma. Please help me find out which one it ...
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1answer
34 views

Borel-Cantelli Lemma Proof.

I struggle to understand the transition between the steps in the red box. Especially why the limit there, i get the intuition behind the limit, its because $A_n$ is a decreasing set ie $ ...
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1answer
28 views

Inclusions regarding the limsup and liminf of sets: $ \liminf E_n \subset \limsup E_n $ [duplicate]

Let $\{ E_n \}_{n \in \mathbb{N} }$ be a sequence of sets in some ambient set $\Omega $. I want to show that $$ \liminf E_n \subset \limsup E_n $$ My attempt: IF $x \in \liminf E_n = ...
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1answer
22 views

trying to prove the characteristic function commutes with limsup

Let $(A_n)$ be a seqeunce of sets, I am trying to show that $$ \limsup_{n \to \infty} \chi_{A_n} = \chi_{\limsup_{n \to \infty}A_n}$$ $$ \liminf_{n \to \infty} \chi_{A_n} = \chi_{\liminf_{n \to ...
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29 views

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 $k>0$, ...
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54 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
64 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
45 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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78 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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40 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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33 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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101 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
28 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
31 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
60 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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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
56 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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45 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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29 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
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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
15 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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61 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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30 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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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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17 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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33 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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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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1answer
36 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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34 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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32 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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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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28 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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92 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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88 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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29 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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38 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
32 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
48 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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90 views

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
37 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
28 views

Measure on intersections of unions

Let $(X,\mathcal{A},μ)$ a measurable space and let $A_1,A_2,...∈\mathcal{A}$, assume that $\sum\limits_{j=1}^{\infty}=\mu (A_j)<\infty$ We have ...