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

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1answer
19 views

Prove that $\liminf_{n\rightarrow\infty}s_n\leqslant \liminf_{n\rightarrow\infty}\sigma_n$ for $\sigma_n=n^{-1}(s_1+…+s_n)$.

For my math class I have to prove that $\liminf_{n\rightarrow\infty}s_n\leqslant \liminf_{n\rightarrow\infty}\sigma_n$ and ...
0
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0answers
17 views

How to prove the following equivalence

$T$ is a positive real number $\lambda_k$ is a family of positive reals number that $\to \infty$ with $k$ $f(s)$ is a function that is a $o(s)$ and $0<f(s)<s$ I want to prove that $\liminf_k ...
0
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0answers
10 views

Recovery sequence for semicontinuous functions

I have seen that the next statement holds (if my memory is not wrong) in a certain book. (I forgot which book this is.) For a lower semicontinuous function $f:(0,1)\to\mathbb{R}$ and a given ...
0
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0answers
32 views

Is it ok to use $x$ and $-x$ as counterexample for $L_{f+g}(M)\le L_f(M)+L_g(M)$?

Let $P$ be partition of $[a,b]$. How to give an counterexample to $L_{f+g}(P)\le L_f(P)+L_g(P)$ and $U_{f+g}(P)\ge U_f(P)+U_g(P)$? Use $f(x)=x$ and $g(x)=-x$ and restricting domain to $(0, 1]$. In ...
-1
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1answer
30 views

Show that $a = \limsup_{n\to\infty} a_n$. [duplicate]

Suppose $a \in \mathbb{R}$ is such that: given any $ε>0$ there exists $n_0 \in \mathbb{N}$ such that $a_n \le a+\varepsilon$ for all $n \ge n_0$ there is $k\ge n_0$ for which ...
0
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1answer
28 views

Proving $\lim \sup (c a_n) = c \cdot \lim \sup a_n$

Let $(a_n)$ be a bounded sequence. Let $c \in \mathbb{R}$ and suppose that $c \geq 0$. Prove then that $$ \lim_{n \to \infty} \sup (ca_n) = c \cdot \lim_{n \to \infty} \sup a_n. $$ Attempt at proof: ...
0
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1answer
28 views

Calculating a Hausdorff Dimension from Formula

I am having difficulty computing the following formula for the Hausdorff Dimension for a type of Moran set called a partial homogenous Cantor set. ...
0
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1answer
23 views

Limsup of a sequence is greater then the limsup of a subsequence? [duplicate]

Consider a sequence of real numbers $\{a_n\}_n$. Let $\{a_{n_j}\}_j$ be a subsequence of $\{a_n\}_n$. Suppose I have shown that $\limsup_{j \rightarrow \infty} a_{n_j}=L$ with $|L|<\infty$. Is it ...
0
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0answers
35 views

Show that $\limsup_{n \to \infty}a_nb_n= \limsup_{n \to \infty}a_n \cdot \lim_{n \to \infty}b_n$ [duplicate]

Let $\{a_n\}_{n \geq0}$ and $\{b_n\}_{n \geq0}$ be two real sequences and suppose that $\lim_{n \to \infty} b_n=b \geq 0$ exist. Show that $$\limsup_{n \to \infty}a_nb_n= \limsup_{n \to \infty}a_n ...
1
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1answer
45 views

Probability of lim sup, lim inf for sequence of random variables.

Maybe this is extremely simple, but i havent found a specific answer for this online. For a sequence of independent continuous random variables $X_n$ ,$n=1,2,3,...$ , all with the same probability ...
0
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0answers
65 views

Limit superior and inferior, giving an example to show that it is possible to have ($a_n$) $>$ $L$ for infinitely many $n$

Question: Suppose that ($a_n$) is a sequence such that L = lim sup ($a_n$) is a real number. Then we know that, for any number $M$ $>$ $L$, there are only finitely many integers $n$ for which ...
1
vote
1answer
30 views

How to prove the limit inferior of a bounded squence exists?

Let $\{a_k\}$ be a bounded sequence of real numbers. Define a sequence $b_k =$inf$ \{a_l|l\ge k \}$ for $k\ge1$. Prove that $\lim_{k\to \infty }$$b_k$ exists. I am proving that the limit inferior ...
2
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2answers
82 views

How can we solve this limit: $\lim _{x\to \infty \:}\left(\frac{\tan\left(x\right)}{x}\right)$?

The limit was $\lim _{x\to \infty \:}\left(\frac{\tan\left(x\right)}{x}\right)$ I tried several ways to solved it, but I didn't find the answer. I think that it doesn't have a limit, but I want to see ...
-1
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1answer
27 views

$\limsup$ and sequence divergence

I don't know how to solve the following question, your help is very much appreciated: Let $(a_n)$ be a sequence where $1\le (a_n) \le 2$ for all $n$. Prove or disprove: if $(a_n)$ is ...
2
votes
1answer
19 views

Limsup, liminf, closure and interior of a Borel set

Consider a sequence of random variables $\{X_n\}_n$ defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$, $X_n:\Omega \rightarrow \mathbb{R}^k$. Let $B\subseteq ...
0
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1answer
27 views

Further clarification on the connection between limsup and liminf for sequences of sets and real numbers

Big picture goal: I am trying to reconcile the difference between between the definition of $\liminf,\limsup$ of a sequence of real numbers and a sequence of sets for the special case of the discrete ...
0
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1answer
32 views

Find limsup and liminf without considering subsequences

I am trying to review some ideas form the calculus course and came across the much hated (back then) topic of $\limsup$, $\liminf$, $\sup$ and $\inf$. I tried to solve the following problem, Find ...
7
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1answer
89 views

A.s. equality between limsup of random variables

"Let $(X_n)_{n\ge 1}$ be a sequence of uniformly bounded random variables defined on a probability space $(\Omega, \mathscr{F}, P)$. Moreover define $\mathscr{F_0}=\{\emptyset,\Omega\}$ and ...
0
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0answers
66 views

extract a subsequence from $\limsup$ convergence

Let $I_\delta:=(-\delta,\delta)$ where $\delta>0$. Let $\{v_\epsilon(x)\}_{\epsilon>0} \subset W^{1,2}(-1,1)$ be a sequence such that $0\leq v_\epsilon\leq 1$ and $v_\epsilon(x)\to1$ a.e. (you ...
0
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0answers
34 views

How to prove this inequality? (limit superior)

How to prove this: $\begin{align} & \underset{\varepsilon \to 0}{\mathop{\lim }}\,{\int_{\begin{smallmatrix} \left( {{t}_{1}},\ \ldots ,\ {{t}_{n}}\ \right)\ \in {{\mathbb{R}}_{+}}.A \\ ...
3
votes
2answers
118 views

In Borel-Cantelli lemma, what is the limit superior?

In a proof of the Borel-Cantelli lemma in the stochastic process textbook, the author used the following. $$\limsup_{n\to\infty}A_n=\bigcap_{n\ge1}\bigcup_{k\ge n} A_k$$ Can someone explain why lim ...
1
vote
1answer
32 views

Obtain the one-side limit by $\liminf$.

Let $I:=(-1,1)$ and function $u$ is defined on $I$. Assume function $u$ is continuous on $(-1,0)$ and $(0,1)$, and we define $u^-(0)=\lim_{x\to 0^-}u(x)$ and $u^+(0)=\lim_{x\to 0^+}u(x)$. We also ...
3
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2answers
40 views

Weakly convergence but not strongly - properties of limsup and liminf

Let $X$ be a Banach space and suppose we have a sequence $\{x_n\}$ which is convergent weakly but not strongly. Define $y_n:=\sum\limits_{k=1}^{n}x_k$. What we can say about ...
0
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2answers
50 views

limsup of sets - proof

Suppose I have a sequence of sets $A_n$. How to prove that elements of the set $\limsup A_n$ defined as $$\limsup A_n = \bigcap_{N=1}^\infty \left( \bigcup_{n\ge N} A_n \right)$$ appear in infinitely ...
3
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1answer
19 views

From “$\text{lim inf}_{k\rightarrow\infty} \|x(k)\|=0 $” to “$\text{lim}_{k\rightarrow\infty} \|x(k)\|=0 $”

Consider the following: If $\ \ \text{lim inf}_{k\rightarrow\infty} \|x(k)\|=0 \ \ $ and $\ \ \text{lim}_{k\rightarrow\infty} \|x(k)\|^2=0\ \ $ then $\ \ \text{lim}_{k\rightarrow\infty} ...
2
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1answer
24 views

Fatou's Lemma: How to apply it for $E[|X|] \leq \liminf_n E[|X_n|]$

Let $X_n$ for $n \geq 0$ be a sequence of random variables with $\sup_n E[X_n^+] < \infty$ and $X_n \overset{P-a.s.}{\rightarrow} X$. We have shown that $$E[X_0] \leq E[X_n ] ...
1
vote
1answer
30 views

Why does this proof about random variable convergence in probability contain $\limsup$ and not $\lim$?

I found this proof on the Internet, but from what I know about convergence in probability, it seems like a $\lim $ would be enough. Am I right, or am I missing something?
2
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1answer
40 views

non-negative almost surely [closed]

I have a probability measure P and a non-negative sequence of random variables $(X_n)$ and the limit $X=\lim X_n$ exists P-almost surely. I would like to show that $X\ge0$ P-almost surely.
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2answers
67 views

Borel-Cantelli lemma and set-theoretic limits

Let $A_n$ be a sequence of events in probability space. The event $A(i.o.)$=$\{A_n$ happens infinitely often $\}$ is formally stated as $\lim \sup A_n$ (in other words $\cap_{k=1}^\infty ...
0
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1answer
28 views

liminf and limsup of events: complement

Consider a sequence $A_i, i \geq 1$ of Events. At one point in our lecture, we have used: $$P[\limsup_i A_i]= 1 -P[\liminf_i A_i^c]$$ with the reasoning that we can use de Morgan's rule. At some ...
4
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2answers
48 views

Limsup of a probability

I've encountered the following questions. Suppose $X_n \rightarrow X$ in distribution, and $a<b$ Prove that $$ P( a \le X\le b) \ge \limsup_{n\to \infty}P(a\le X_n\le b) $$ I know how to find ...
2
votes
1answer
70 views

limit supremum that I have not seen before - for $t\to0^+$

Suppose I have a function $f(t) = 0$ for $t \leq 0$ and $f(t) = 2t\sin(1/t) - \cos(1/t)$ for $t > 0$. I want to show that $f$ is discontinuous at $0$ by showing that $\limsup_{t \to 0^+} f(t) = 1$ ...
0
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2answers
25 views

Problem with $\mathbb{P}(\liminf_n(A_n\cup B_n))$

We know that $\mathbb{P}(\liminf_n A_n)=0.3$ and $\mathbb{P}(\limsup_n B_n)=0$. Find $\mathbb{P}(\liminf_n(A_n\cup B_n))$. My solution: We know that $\liminf_n(A_n\cup B_n)\supset \liminf_nA_n \cup ...
1
vote
1answer
29 views

Liminf and limsup of subsequences

Consider a bounded sequence $\{A_n\}_n$ and a subsequence $\{A_{n_k}\}_k \subseteq \{A_n\}_n$. Is it true that $$ \liminf_{n\rightarrow \infty}A_n \geq \liminf_{k\rightarrow \infty}A_{n_k} $$ and $$ ...
0
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0answers
69 views

Proving if $\lim|s_n|=0$ then $\limsup|s_n|=0$.

I am trying to prove that if $\lim|s_n|=0$ then $\limsup|s_n|=0$. My proof goes as follows: Suppose $\lim_{n \rightarrow \infty} s_n=0$, then this implies that $\forall \epsilon>0,\ \exists N$ ...
0
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1answer
19 views

finding the limit superior and inferior of the sequence $\frac 1 n \cos{\frac{n\pi}{2}}$

I'm doing a problem in which I should find the limit superior and inferior of the sequence $\left\{ \frac 1 n \cos{\frac{n\pi}{2}} \right\}^\infty_{n=1}$ If $n$ is even, then ...
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2answers
57 views

Suppose $ \lim \left( a_{n}+a_{n+1} \right)=0 $. Show that $ \lim a_{n}=0 $ or $ 0 < \limsup a_{n} $.

I tried separating into cases, where $ a_{n} $ converges (thus is bounded), is unbounded (thus diverges) and is bounded and also diverges. I also tried proving by way of contradiction. I am having ...
1
vote
1answer
33 views

If $\limsup|a_n|^{\frac{1}{n}} = \frac{1}{R}$ and $\frac{1}{r} > \frac{1}{R}$, why is there an $N$ such that $|a_n|^{1/n}<1/r$ for all $n\ge N$?

I'm reading Conway's Functions of One Complex Variable and I didn't understand this proof on page $31$: I didn't understand why $\frac{1}{r}>\frac{1}{R}$ implies there is an integer $N$ such ...
2
votes
1answer
62 views

If $\limsup_{n\to\infty} \ x_{n} = a$ then why does it exist a subsquence $s_{n}$, which $\lim_{n\to\infty} \ s_{n} = a$?

Maybe it seems trivial since $\limsup$ is known as the "greatest limit point of $x_n$", so there's a subsequence which converges to $a$. But I cannot use this definition. Is it possible to prove it ...
1
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1answer
31 views

reference on limsup and liminf for functions

I need a reference well-explaining (definitions and useful properties) the notions of upper limit and lower limit for functions defined on a topological space Thank you.
0
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0answers
9 views

Set of limit of sub sequence for this sequence

Which of following options is set of limit of sub sequence for this $a_n= \frac{n}{e} -[ \frac {n}{e}]$ 1) [0,1] 2)(0,1) 3) $ \emptyset$ 4){0} We know that set of limit of sub sequence is ...
0
votes
1answer
26 views

If $\lim_{x \to \infty} x_n = \liminf_{n \to \infty} x_n = \limsup_{n \to \infty} x_n= -\infty$ does it exist a convergent subsequence of $x_n$?

If $\lim_{x \to \infty} x_n = \liminf_{n \to \infty} x_n = \limsup_{n \to \infty} x_n= -\infty$ does it exist a convergent subsequence of $x_n$? I've learned that the limit of a convergent subsequence ...
0
votes
2answers
38 views

If $\limsup_{n\to\infty} \ x_{n} =-\infty$ then why $\lim_{n\to\infty} \ x_{n} =-\infty$?

I cannot understand why, if $\limsup_{n\to\infty} \ x_{n} =-\infty$ then $\lim_{n\to\infty} \ x_{n} =-\infty$? Can anybody explain it? What's the relationship between $\limsup$ and $\lim$?
0
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2answers
31 views

Can we always find the infimum and supremum of a sequence?

I've found in my book that: $$\liminf_{n\to\infty} \ x_{n} = \sup\{\inf\{x_{k}:k\geq n \}:n \in \mathbb{N}\}$$ $$\limsup_{n\to\infty} \ x_{n} = \inf\{\sup\{x_{k}:k\geq n \}:n \in \mathbb{N}\}$$ If ...
0
votes
1answer
41 views

Consequences of $\limsup\limits_{n \rightarrow \infty}(a_n)=1$.

$\{{a_n}\}$ is a sequence with $\limsup\limits_{n \rightarrow \infty}(a_n)=1$. I have trouble with these, so I want to verify these (a) $a_n \leq1$ eventually; False (b) $a_n \leq2$ eventually; ...
1
vote
1answer
37 views

Finding lim sup and lim inf

Given the sequence $$ (a_n)=\begin{cases} 3^{-n}, & \text{for even }n \\ 5^{-n}, & \text{for odd } n \end{cases} $$ How to find: $$\limsup_{n\to\infty}\frac{a_{n+1}}{a_n}$$ ...
1
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0answers
30 views

How to find $\sup \Pi_{i=0}^{n} (\sin(i)^2 - \frac{25}{16})$?

Let $\sup,\inf,{\rm dif}$ denote resp supremum , infimum and $\rm dif$ = supremum - infimum. Does any of the 3 below have a closed form ? $\sup \Pi_{i=0}^{n} (\sin(i)^2 - \frac{25}{16})$ $\inf ...
1
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0answers
17 views

If $a_n \geq b_n$ and $b_n$ converges to $b$ then $\liminf a_n\geq b$ [duplicate]

Suppose $\{a_n\}$, $\{b_n\}$ are sequences in $\mathbb R$. Suppose $a_n\geq b_n$ and $\{b_n\}$ converges to $b$. Can we conclude that $\liminf a_n\geq b$. What if we further assume $\{a_n\}$ is ...
1
vote
0answers
38 views

Sketch of a possible equivalence with Riemann hypothesis

From Robin's equivalence (see [1]) and the following trigonometrics identitites, I ask to me if it is feasible write vagues equivalences using this strong result and if these equivalences will be ...
4
votes
1answer
70 views

Almost Surely convergence with Bernoulli

How can I demonstrate that a sequence of Bernoulli Random Variables Xn with parameter $\frac1{2n^2}$ converges almost surely to some random variable X? I know that $X_n$ takes the value $1$ with ...