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

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Question of double limits

I post a problem concerning a possible generalization of the question of interchanging in double limits. Given a sequence of functions $\{f_{j}\}_{j}$ on an interval $I$ and a point $a\in I$, is it ...
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1answer
36 views

IF $\sum_{n=1}^{\infty}P(|X_n|>c)=\infty\forall c>0$, then $\limsup_{n\rightarrow\infty}|S_n|=\infty$ a.s.

If $S_n=\sum_{i=1}^n X_i$ where $X_i,i\ge 1$ are iid and $\sum_{n=1}^{\infty}P(|X_n|>c)=\infty\forall c>0$, then $\limsup_{n\rightarrow\infty}|S_n|=\infty$ a.s. I tried this but got stuck: Let ...
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1answer
54 views

Is this an increasing sequence?

From this beginning of this document - Let $(c_n)$ be a bounded sequence of real numbers. Define $a_n = \inf\{c_k : k ≥ n\}$ The sequence $(a_n)$ is bounded and increasing so it has a ...
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1answer
36 views

Trying to prove lim inf $(A_n) \subseteq$ lim sup $(A_n)$

I am trying to show that for a sequence of sets $(A_n)$ $$ lim \ inf \ (A_n) \subseteq lim \ sup \ (A_n)$$ I can see why this is true intuitively as follows - $x \in lim \ inf \ (A_n)$ $\implies ...
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1answer
29 views

Applying limit inferior

In the solution to this question while attempting to prove by contradiction it states that if we assume $|x(t)|\nrightarrow \infty $ then $M:=$lim inf $_{t\uparrow b}|x(t)|$ Why can I just not use ...
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3answers
84 views

Proving $\limsup\frac 1 {a_n}=\frac 1 {\liminf a_n}$ and $\limsup a_n\cdot \limsup \frac 1 {a_n} \ge 1$

Let $a_n$ be a sequence such that $\forall n\in \mathbb n: 0<a\le a_n\le b <\infty.$ Prove: $\displaystyle\limsup_{n\to\infty}\frac 1 {a_n}=\frac 1 ...
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2answers
61 views

Questions on limit superiors

Given any bounded sequence $x(n)$, let $l$ be its limit superior. Let $\epsilon > 0$. Does there exist $n \in \mathbb{N}$ such that $x(n)<l+\epsilon$? Does there exist infinitely many $n$ such ...
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0answers
43 views

Limit of function Problem

Can anybody send me proof for the following problem? Let $f(x)$ be a function continuous on closed interval $[a, b]$. Prove that $$ \lim_{\underset{c \in (a,b)}{c\to 0}} \inf (f (\{ x \mid b - c ...
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18 views

Show that if $\lim_{k \rightarrow \infty} \sup (\sqrt[k]{|a_k|}) < 1$ then $\sum_{k=0}^\infty a_k$ converges absolutely.

Let $(a_k)_{k \in \mathbb{N}}$ be some real sequence. Show that if $\lim_{k \rightarrow \infty} \sup (\sqrt[k]{|a_k|}) < 1$ then $\sum_{k=0}^\infty a_k$ converges absolutely. I have some general ...
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1answer
27 views

Show that if $(a_k)_{k \in \mathbb{N}}, (b_k)_{k \in \mathbb{N}}$ are bounded, then $\lim \sup(a_k + b_k) \leq \lim \sup(a_k)+\lim \sup (b_k)$

Show that if $(a_k)_{k \in \mathbb{N}}, (b_k)_{k \in \mathbb{N}}$ are bounded, then $\lim \sup(a_k + b_k) \leq \lim \sup(a_k)+\lim \sup (b_k)$ I'm not quite sure how to prove this. I want to try ...
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2answers
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Find $\lim_{k \rightarrow \infty} \sup a_k$ and $\lim_{k \rightarrow \infty} \inf (a_k)$, with $a_k=(\frac{1}{k})_{k \in \mathbb{N}}$

Find $\lim_{k \rightarrow \infty} \sup a_k$ and $\lim_{k \rightarrow \infty} \inf (a_k)$, with $a_k=(\frac{1}{k})$ Per definition: $\lim_{k \rightarrow \infty} \sup(a_k) = \lim_{k \rightarrow \infty} ...
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2answers
59 views

When can I interchange $\liminf$ and integral?

Let $g_n:[0,T] \to \mathbb{R}$ be a sequence. When can I say that $$\int_0^T \liminf_{n \to \infty} g_n(t) \leq \liminf_{n \to \infty}\int_0^T g_n(t)?$$ Under what circumstances?
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2answers
35 views

Help with understanding a proof about differentiating a real power series

I'm stuck trying to understand a proof of the following theorem: Let $ \sum a_nx^n$ be a power series with radius of convergence $ R $. Then $ \sum na_nx^{n-1}$ also has radius of convergence $ R $. ...
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1answer
23 views

Complement of limsup E_n

Im reading Friedmans modern analysis, and im having some flaws in my logical thinking... the question is to prove that $\left(\limsup\limits_{n}E_n\right)^c = \liminf\limits_{n}E_n^c$. My first ...
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2answers
307 views

$\int_0^1f(x)dx=1, \int_0^1xf(x)dx=\frac16$ minimum value of $\int_0^1f^2(x) dx$?

Let $f(x)\geq0$ be a Riemann integrable function, and $$\int_0^1f(x)\,\mathrm dx=1, \int_0^1xf(x)\,\mathrm dx=\frac16.$$ Find the minimum value of $\int_0^1f^2(x)\,\mathrm dx$ Cauchy-Schwarz ...
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1answer
65 views

A couple of inequalities (explanation needed), how to show $\liminf_{k \to 0}k^{-1}\int(\nabla u - \nabla v)\nabla (T_k(b(u)-b(v))) \geq 0$

Let $T_k(x) = \max\{-k, \min(x,k)\}$, a truncation function at levels $k$ and $-k$. Let $b$ be a Lipschitz increasing function with $b(0)=0$ and $b'$ and $b^{-1}$ also Lipschitz. I have seen it ...
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1answer
34 views

Prove $\{Z=0\}\subset\limsup\limits_{n}\{X_n<\epsilon\}$

Let $(X_n)_{n\in\mathbb N}$ be independent real random variables, with values in $(0,\infty)$. Consider the random variable $Z(w):=\inf\limits_{n\in\mathbb N}X_n(w)$. Prove that for every fixed ...
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1answer
71 views

Show that $\limsup_{n\rightarrow\infty} \|x_n\|^{\frac{1}{n}}=1$

Let $(x_n)$ be weakly convergent, but not norm convergent, sequence from a Banach space. Show that $\limsup_{n\rightarrow\infty} \|x_n\|^{\frac{1}{n}}=1$. Any help?
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1answer
26 views

Interchaging $P(\mathrm{limsup})$ with $P(\mathrm{limit})$ for $P$ a probability measure.

I have been going through Resnick's 'A Probability Path', and at one point he is trying to prove a version of Fatou's lemma: $$P(\liminf_{n\rightarrow\infty}A_n)\le\liminf_{n\rightarrow\infty}P(A_n)$$ ...
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1answer
52 views

Definition of $\limsup$

please tel me what is the definition of $$\limsup_{|u|\rightarrow\infty}\frac{2F(t,u)}{|u|^2}<\lambda$$ using $\varepsilon$ Please Thank you
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36 views

Question concerning $\limsup$

I have this two hypothesis where $q\geq 1$ and where $F(x,t)=\int_0^t f(x,s)ds$, p=2 I dont understand how they find this (3.5) Please help me thank you
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1answer
26 views

Limit of a sequence of roots

Let $L$ be a real number and $\rho \in (0, 1)$. Define the following sequence, $a_0 = 0$, $a_1 = L^\rho$, $a_i = (L + a_{i-1})^{\rho}$. Is $\lim\limits_{i \rightarrow \infty} a_i= \infty$ for all ...
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1answer
58 views

What must a function satisfy in order to say that a certain limit exists?

Suppose that $f:R_+\to R_+$ is $C^2$, and that $f(0)=f'(0)=f''(0)=0$, and that for all $x$, $\frac{xf''(x)}{f'(x)}\geq1$. From here we can safely say that $\lim\inf_{x\to0}\frac{xf''(x)}{f'(x)}\geq1$. ...
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1answer
30 views

updated: lim inf, lim sup and bertrand's postulate for a function

Define $q:\mathbb{N}\rightarrow\mathbb{Q}^+$ with, $q(n)=min\{\alpha\in \mathbb{Q}^+\mid \exists \;p\in \mathbb{N},\; p \text{ prime and } p \in (n,\alpha n]\}$. Is the ...
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1answer
30 views

Complex analysis problem, using limsup

Im trying to solve the following problem. Let $\Omega \in \mathbb{C}$ an open bounded set, let $f\colon \Omega \to \mathbb{C}$ be holomorphic, and supose there exists a constant $M \geq 0$ wich ...
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0answers
28 views

$\limsup \sqrt[n]{(\lambda a_n)} \leq \limsup \sqrt[n]{b_n}$ given pointwise inequality

Let $(a_n),(b_n)$ be non-negative real sequences, $0 < \lambda \in \mathbb{R}$ and $m \in \mathbb{N}$. Further assume $\lambda^\frac{1}{n}a_n^\frac{1}{n} \leq ...
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2answers
77 views

What does $\liminf_{n\to \infty} f_n$ and $\limsup_{n\to \infty} f_n$ mean?

I have searched the internet, including this website, and I have found some answers, but none which I understand what actually mean. What is $\liminf_{n\to \infty} f_n$? Is it a single value? Is it a ...
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1answer
92 views

Characterization of lim sup, lim inf

If $(a_n)$ is a real sequence, in lecture we had: $\limsup_{n\to\infty} a_n=a \iff (i)\forall \epsilon >0 \,\exists n_0\in \mathbb{N} :a_n<a+\epsilon$ $ \forall n\ge n_0$ and $(ii) \forall ...
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24 views

Determining the expression of supremum correctly - $\sup_{x\in [0,1]} |f-f_{n}|$.

Let $f_{n}$ be a sequence of functions and let $f=\lim_{n\to\infty}f_{n}$. If $f_{n}=nx^{n}(1-x)$ then $f=0$. How do I determine the following expression $$ \sup_{x\in [0,1]} |f-f_{n}|=\sup_{x\in ...
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1answer
42 views

Product of Limitsuperior of bounded sequences

$\{a_{n}\}$,$\{b_{n}\}$ are two bounded sequences.How can we prove that $\limsup(a_{n}b_{n})$ = $\lim(a_{n})\limsup(b_{n})$. Is it equal to $\lim(a_{n})\lim(b_{n})$? If the sequences are not ...
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1answer
44 views

$\limsup s_n = \infty$, $\liminf t_n >0$, prove that $\limsup s_n t_n = \infty$

I need some help with the last step. Here's what I have already done. Proof: $\limsup s_n =0$ implies that $\lim_{N\to\infty}\sup\{s_n:n>N\}=\infty$, by definition. We know that ...
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1answer
39 views

Limit superior and limit of a sequence

$\{x_n\}$ be a bounded sequence such that for every bounded function $\{y_n\}$, $$\limsup\{x_{n}+y_{n}\} =\limsup\{x_{n}\}+\limsup\{y_{n}\}$$ then prove that {$x_{n}$} is convergent. This is a ...
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2answers
32 views

Counterexample for limsup

Statement: $\limsup c_n a_n = c \limsup a_n$ Please help find a counterexample to this statement if $c<0$. Edit: also suppose $c_n$ -> c and $\limsup a_n$ is finite
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2answers
37 views

Prove that limsup is the supremum of the limit points

Let ${a_n}$ be a bounded sequence of real numbers, and let P be the set of limit points. Prove that $\limsup a_n = \sup P$. I have proved that there must be a subsequence that converges to ...
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1answer
20 views

Is this limsup calculation correct?

$a_n=1$ if $n=2^k$ for $k>0$ $a_n=\frac{1}{n!}$ otherwise a) Find limsup $\displaystyle \frac{|a_{n+1}|}{|a_n|}$ - I think this is infinity because we can find a term that is 1/something!, but ...
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1answer
25 views

Product of lim sups

Suppose lim sup $a_n$ is finite, and $c_n -> c$ Prove that if $c \geq 0$ lim sup $a_n c_n$ = c lim sum $a_n$ and find a counterexample to this if $c <0$. Is there a rule that the product of ...
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1answer
33 views

limsup of series

Find limsup $|a_j|^{1/j}$ $\displaystyle a_j = \sum_{j=1}^\infty \frac{(1+1/j)^{2j}}{e^j}$ Since the limit of the numerator is $e^2$, is it correct that the limsup is equal to 0? How can I write out ...
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1answer
48 views

relations between the root test and the ratio test

relations between the root test and the ratio test I know the theorem is correct if they are exist $$ \lim\inf\limits_{n\rightarrow \infty} \frac{A_{n+1}}{A_n} \leq \lim\inf\limits_{n\rightarrow ...
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49 views

Show that $\liminf\limits_{n\rightarrow\infty}a_{n}/a_{n-k}\leq \liminf\limits_{n\rightarrow\infty}a_{n}^{k/n}$

Here is an exercise: Let $\{a_{n}\}$ be a positive increasing sequence, can we prove that: $\liminf_{n\rightarrow\infty}\frac{a_{n}}{a_{n-k}}\leq \liminf_{n\rightarrow\infty}a_{n}^{k/n}$? Could ...
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1answer
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radius of symmetric random walk on $\mathbb{Z}$

How to calculate the radius of the symmetric random walk on $\mathbb{Z}$, i.e. $\limsup_k (p^{(k)}(0,0))^\frac{1}{k}$? ($p^{(k)}(0,0)$ denotes the probability to get from $0$ to $0$ in $k$ steps and ...
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1answer
28 views

How to show that the limit of a function does not exist, while $\limsup$ and $\liminf$ do?

It does seem like a very simple question, but I don't see how to do it properly. How can we show that the limit of $$3^{\lfloor\frac{-\log(j)}{\log(2)}\rfloor}j^{\frac{\log(3)}{\log(2)}}$$ as ...
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3answers
46 views

Lim sup definition doesn't make sense to me

I don't really understand the concept of Lim Sup and why it is any different from a standard limit. I am working on a question about radius of convergence and I know that $r = ...
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1answer
24 views

Derivatives from the left and right

I need to give a definition of the derivative from right and from left. (I know it doesn't make a whole lot of sense, but it is supposed to be similar to the same thing as the limits from the right ...
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1answer
72 views

measure space problem.

Let $(\Omega,\mathcal{F},\mu) $ be a probability space. Let $\delta>0$ and for each $n\in \mathbb{N}$. Let $A_n \in \mathcal{F}$ satisfy $\mu(A_n)\ge\delta$. Prove that the set $A_\infty $ ...
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1answer
42 views

What are the lim sups and lim infs of a sequence?

What relation, if any, can you state for the lim sups and lim infs of a sequence {an} and one of its subsequences {$a_n$$_k$}?
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5answers
400 views

Understanding limsup

My textbook says: $$\overline{s}_n = \sup \{a_n \mid n \geq N\}$$ and $\operatorname{limsup} \{a_n\}_{n \to \infty} = \lim_{N \to \infty} \overline{s}_N$. Also, it says: As $N$ gets larger, the sup ...
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2answers
56 views

$\limsup \sqrt[n]{a_n} \leq \limsup a_n$ if $(a_n)$ is non-negative sequence?

Let $(a_n)$ be a sequence with $0 \leq a_n \leq 1$. Is it possible to show, that $\limsup \sqrt[n]{a_n} \leq \limsup a_n$? Conversely, if $(a_n)$ is a sequence with $0 \leq a_n$ and $\limsup ...
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0answers
35 views

Stochastic big O notation

Let $||.||$ indicate the Euclidean norm. Let $\theta_0$ be a specific value of the parameter $\theta \in \Theta\subseteq \mathbb{R}^d$ and $G_n$ be a random vector-valued function $$ G_n : \Theta ...
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1answer
48 views

Help proving a limit exists using limsup and liminf.

Let $X=(x_n)_{n\in N}$ be a bounded sequence of real numbers with $\liminf_{n \to \infty}x_n=\limsup_{n \to \infty}x_n$. Prove that $X$ is convergent. I'm not sure where to start. Any suggestions?
2
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1answer
83 views

Proving the Kochen-Stone lemma using the Paley-Zygmund inequality

I am trying to understand a proof to a lemma by Kochen and Stone which appears here, using the Paley-Zygmund inequality. I'll repeat the proof in a detailed manner, and explain what bothers me about ...