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

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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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33 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
34 views

Sup and inf of bounded sequence [on hold]

Let $(x_n)$ be a bounded sequence, let $L$ be the set of all accumulation points of $(x_n)$, and $S=\inf{\{\sup{\{x_k:k\geq n\}}\}}$, prove that $S=\sup{L}$
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26 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
28 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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0answers
20 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
28 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
50 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 ...
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1answer
16 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
35 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
73 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
23 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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0answers
17 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 ...
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2answers
30 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
19 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
44 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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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 ...
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1answer
23 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
29 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
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1answer
27 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
66 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
45 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
28 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 ...
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1answer
61 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
31 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 ...
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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 : ...
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2answers
84 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
80 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 ...
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0answers
66 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
45 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 ...
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2answers
58 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&= ...
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2answers
60 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 ...
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0answers
22 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 ...
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1answer
32 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 ...
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1answer
40 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 ...
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2answers
64 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 $} ...
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1answer
48 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
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1answer
55 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 ...
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1answer
36 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 ...
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2answers
58 views

For random variables, show that $\limsup\limits_nX_n<\infty\Longrightarrow \sup_n X_n<\infty$

Why is the following true ? $$\limsup\limits_nX_n<\infty\Longrightarrow \sup_n X_n<\infty$$ where, $X_n's$ are random variables. If we consider only finitely many $X_n$, say ...
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1answer
35 views

Prove $\liminf(a_n + b_n) \le a + B$ (Using “$\varepsilon$ language”)

Denote: $\liminf a_n = a$ and $\limsup b_n = B$. Prove: $\liminf(a_n + b_n) \le a+B$. The proof: Let $\varepsilon > 0$. By definition of infimum, there's a subsequence $a_{n_k}$ such that ...
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1answer
59 views

Is liminf of a product equal to the product of liminfs?

My question is just for curiosity. I was thinking if is true this curious affirmation: Let $a_n$ a bounded sequence of nonnegative numbers and $b_n$ a convergent sequence of negative numbers. Then ...
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2answers
54 views

Sandwiching Limsups & liminfs of expectations

Why is it that if we sandwich a liminf of an expectation between two equal quantities we get that the limit exists? Can we somehow deduce the limsup from that and conclude that it's the same or am I ...
4
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1answer
55 views

Bounded limit inferior and limit superior of functions on dense sets implies divergence on uncountable dense set.

Reading an article by Akcoglu et al. "The strong sweeping out property for lacunary sequences, Riemann sums, convolution powers, and related matters", which can be found here: ...
1
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1answer
26 views

Coin tossing, two heads always followed by two tails - lim sup necessary?

In Bernoulli Space $\Omega$, let $E_n$ be the event that the $n$th toss is heads. Write down a formula in terms of the $E_n$ for the following event: “Every time two Heads appear in succession, the ...
3
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0answers
68 views

Question about the definition of $\limsup$ and $\liminf$ on real valued functions

I understood the definition of $\liminf$ and $\limsup$ for sequences well. But there is a little bit of confusion when it comes to functions. If I did learn it correctly, the definition of ...
2
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0answers
43 views

when we can say that $\limsup_{n\to\infty}f(a_n)=f(\limsup_{n\to\infty}a_n)$?

when we can say that $\limsup_{n\to\infty}f(a_n)=f(\limsup_{n\to\infty}a_n)$? (What conditions must have the function $f$?)
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1answer
80 views

Problems understanding definition of limit superior.

I'm aware that there are multiple questions on this topic and I have read most of them, but I still don't really understand the definition and would really appreciate some quick help with this, ...
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1answer
33 views

Is my proof about $\liminf$ of bounded real sequences correct?

Yesterday I asked in the question at A proof about the limit infimum of a bounded sequence about the proof of the following statement: Let $x_n$ be a bounded sequence of real numbers. Then the ...