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

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5
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2answers
69 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
46 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
votes
0answers
33 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
38 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
41 views

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

Suppose that ${Bn: n \geq 1}$ 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
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 ...
0
votes
0answers
16 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
20 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
39 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
50 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
45 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
49 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
33 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 ...
0
votes
2answers
55 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 ...
1
vote
1answer
33 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 ...
1
vote
1answer
58 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 ...
2
votes
2answers
46 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
votes
1answer
40 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
vote
1answer
18 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
votes
0answers
62 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
votes
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 $f$?)
1
vote
1answer
64 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, ...
0
votes
1answer
31 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 ...
1
vote
1answer
49 views

A proof about the limit infimum of a bounded sequence

I tried to find a proof for this statement: If we have a bounded sequence $x_n$, then the limit infimum is defined as $a=\liminf_n x_n$ such that $a$ is the largest of real numbers which have the ...
0
votes
2answers
104 views

The definitions of limit infimum and limit supremum

I have begun reading Rosenthal's "A First Look to Rigorous Probability Theory" and in order to reinforce my calculus background I am studying through the appendix section. Here he defined the limit ...
3
votes
0answers
131 views

A limsup version of the Principle of Uniform Boundedness?

Suppose $X$ is a Banach space and let $\{f_\alpha\}$ be a net of continuous linear functionals satisfying $\limsup_{\alpha} | f_\alpha(x) | < \infty$ for each fixed $x \in X$. Is it true that ...
1
vote
1answer
54 views

Subsequences and limit inferior

Suppose $f\colon \mathbb{R} \to \mathbb{R}$ is a continuous function. Let $x\in \mathbb{R}$ and let $(x_n) \subseteq \mathbb{R}$ be a sequence converging to $x$. Let $(y_n)$ be a subsequence of ...
1
vote
1answer
56 views

Limsup of intervals $(-\infty, - n \sin(1/n))$

I think that $\limsup \big (-\infty, - n \sin \frac{1}{n} \big )$ (as the lim sup of sets) is $(-\infty, 1)$. Is this correct? If so, how do we prove it? Background: I'm trying to do a question in ...
1
vote
1answer
51 views

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 ...
0
votes
1answer
38 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 ...
0
votes
1answer
57 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 ...
3
votes
1answer
66 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 ...
1
vote
1answer
31 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 ...
5
votes
3answers
114 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 ...
1
vote
2answers
70 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 ...
0
votes
0answers
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 ...
0
votes
1answer
30 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 ...
2
votes
2answers
16 views

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} ...
0
votes
2answers
61 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?
3
votes
2answers
36 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 $. ...
0
votes
1answer
33 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 ...
18
votes
2answers
336 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 ...
1
vote
1answer
67 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 ...
1
vote
1answer
37 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 ...
1
vote
1answer
72 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?
0
votes
1answer
32 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: $$P(\liminf_{n\rightarrow\infty}A_n)\le\liminf_{n\rightarrow\infty}P(A_n)$$ ...
0
votes
1answer
53 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
0
votes
1answer
28 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 ...
3
votes
1answer
59 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 $\lim\inf_{x\to0}\frac{xf''(x)}{f'(x)}\geq1$. ...
0
votes
1answer
31 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 ...