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

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2
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1answer
38 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 ...
2
votes
1answer
22 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 ...
0
votes
1answer
34 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} ...
0
votes
2answers
37 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 ...
0
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1answer
24 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 $\epsilon$, ...
1
vote
1answer
38 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 ...
1
vote
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 ...
0
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3answers
54 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 = ...
0
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1answer
27 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 ...
0
votes
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 ...
0
votes
0answers
10 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 ...
0
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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 = ...
2
votes
0answers
48 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= ...
0
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0answers
35 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 ...
0
votes
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 ...
0
votes
0answers
31 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}, ...
2
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0answers
30 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 $\frac{1}{r} = \limsup |a_n|^{1/n}$. ...
1
vote
2answers
51 views

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 ...
0
votes
1answer
19 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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0answers
54 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 ...
2
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3answers
82 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 : $$ ...
2
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0answers
26 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 ...
0
votes
0answers
18 views

Meaning of $\liminf \frac{an}{bn} > 0$ and $\limsup \frac{an}{bn} < \infty$

I am having trouble understanding what is meant by $\liminf_{N}\frac{a_{N}}{b_{N}} > 0$ and $\limsup_{N}\frac{a_{N}}{b_{N}} < \infty$ for $N\to \infty$. I think that it has something to do with ...
3
votes
2answers
34 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 $\lim_{n \to \infty} ||x_n|| = ||\lim_{x \to \infty} x_n||$?
1
vote
1answer
25 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 ...
0
votes
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 ...
1
vote
0answers
89 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 ...
0
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1answer
26 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} ...
1
vote
1answer
31 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 ...
2
votes
1answer
29 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 ...
1
vote
2answers
73 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?
1
vote
1answer
52 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 ...
1
vote
0answers
32 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 ...
1
vote
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
vote
1answer
74 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 $\liminf x_n ≤ \liminf y_n$.
0
votes
1answer
37 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
vote
1answer
45 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
votes
2answers
92 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 ...
1
vote
1answer
86 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
87 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
vote
1answer
46 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 ...
1
vote
2answers
68 views

Show that lim inf Bn and lim sup Bn equals to a null set

Suppose that ${B_n: n \geq 1}$ is a sequence of disjoint set. Show that $$\begin{align}\limsup_{n\rightarrow \infty}B_n &= \emptyset \text{ and}\\ \liminf_{n \rightarrow \infty}B_n&= ...
1
vote
2answers
61 views

Limsup, showing that two expressions are equal

I am stuck at this problem which I use for something else. If $\{a_i\}$ is a sequence of number then I want to prove that $\limsup |a_i|^{1/i}=\limsup|a_{i+k}|^{|1/i}$, where k is a fixed positive ...
0
votes
0answers
23 views

Relation between monotonocity and Lim sup of Ratio

Let $x_n$ be a real number sequence. Is it true that : i) if $x_n$ is a bounded eventually monotonicly non-increasing sequence, then lim sup $(x_{n+1}/x_{n}) \leq 1 $. ii) If $x_n$ is a ...
0
votes
1answer
39 views

Lim sup ratio test and Ramifications

Let $x_n$ be a real number sequence. I have managed to prove that,for a sequence $x_n$ of positive terms : if lim sup $(x_{n+1}/x_{n}) < 1$ then $x_n$ is not only eventually monotonicly ...
1
vote
1answer
41 views

For every intermediate value, there exists a sequence that converges to it.

I want to prove that: If the continuous function $f(x)$ has a bounded limt as $x$ goes to $\infty$ i.e $$0<L=\liminf (f)\leq S=\limsup(f)<\infty,$$ then for every $x_0 \in [L,S]$ there ...
3
votes
2answers
69 views

Integrating the two-views of lim sup and lim inf

Preliminaries Let $(x_n)$ be a bounded real number sequence and $ (x_n )_{n≥k} $be a subsequence of $x_n$ which only takes the values of the sequence starting from the k−th term. Let {$x_n $} ...
-1
votes
1answer
49 views

If $f$ is increasing toward $1$, then $\sup\{f(x)\sin x \}=1$

Suppose $f$ is an increasing monotone function in $(0,\infty)$. If $$\lim_{x \to \infty} f(x)=1$$ then $$\sup\{f(x)\sin x\mid x>0\}=1$$ I am not really sure how to approach this, any help will ...
3
votes
1answer
60 views

Convergence of discrete random variables, show $\frac{S_n}{\sqrt{n}}\to0$ a.s.

Let $X_n$ be a sequence of independent discrete real random variables, with discrete density $$p_{X_n}(x):=\Pr(X_n=x)= \cases{ 1-\frac1{n^2} & \text{if } x= 0\cr \frac1{2n^2} & \text{if ...
0
votes
1answer
38 views

Is the set $\limsup\limits_n\{\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$ equal to $\{\limsup\limits_n\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$?

Difference between $\limsup\limits_n\{\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$ and $\{\limsup\limits_n\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$ are the sets equal ? I think they would ...