For questions concerning the definition and properties of limit superior and limit inferior.
4
votes
1answer
68 views
A question on limsup
Let $a_n>0$. Prove that $$\varlimsup_{n\to\infty}n\left(\frac{1+a_{n+1}}{a_n}-1\right)\geq 1.$$
I argue by contradiction. If it is not ture, then $$\exists\ N,\ \forall\ n\geq N, ...
2
votes
1answer
82 views
Borel-Cantelli Lemma
I have some difficulties understanding the following:
Let $(X_n)$ be a sequence of independent random variables s.t.
$P[X_n=1]=1−P[X_n=0]=\frac{1}{n}$
After using the Borell Cantelli lemma, I ...
0
votes
2answers
31 views
Supremum of $\underset{n \to \infty }{\lim} \underset{x\in [0,1]}{\sup} \left | \frac{x+x^{2}}{1+n+x} \right | $
I need to find $$\underset{n \to \infty}{\lim} \underset{x\in [0,1]}{\sup} \left| \frac{x+x^{2}}{1+n+x} \right|.$$ How to show that supremum will be at the point $x=1$?
1
vote
1answer
30 views
Two questions on $\limsup$: do nested ones commute and sending $n$ to $-\infty$
As for the second question, this is really just me wondering on definitions. For a function $f:\mathbb{R}\to\mathbb{R}$ define
$$\limsup_{x\to+\infty}f(x):=\lim_{x\to+\infty}\sup_{z\geq x}f(x).$$
...
1
vote
0answers
48 views
Show that $\liminf a_n \le \liminf s_n$ [duplicate]
Let $a_1, a_2, a_3,\dotsc$ be a bounded sequence of real numbers. Define
$s_n =\frac{(a_1 + a_2 + ...+ a_n)}{n}$, $n \in \mathbb{N}$
Show that $\liminf a_n \le \liminf s_n$
Can I get some help? I ...
3
votes
1answer
49 views
Is it true that $\cup_{n=N}^{\infty}A_{n}\setminus\cap_{n=N}^{\infty}A_{n}=\cup_{n=N}^{\infty}A_{n}\triangle A_{n+1}$?
During an exam I have claimed that if $\{A_{n}\}_{n=1}^{\infty}$
then for any $N\in\mathbb{N}$
$$\limsup A_{n}\setminus\liminf ...
1
vote
1answer
32 views
Suppose that for $a_n\geq b_n$ for all $n$. Show that $\varliminf_{n \to \infty} a_n\geq \varliminf_{n \to \infty} b_n$.
This is what I have so far:
Since $a_n\ge b_n$ for every $n$ then we have that $\inf\{a_n; n\ge k\} \ge \inf\{b_n; n\ge k\}$ for every $n$. When we take the limit as $n\rightarrow \infty$ we get ...
4
votes
1answer
54 views
Sup and inf properties with bounded sets
Let $A, B \subset\mathbb{R}$ be bounded sets. Show $$\sup{A}-\inf{B}=\sup\{a-b:a\in A, b\in B\}$$
2
votes
1answer
28 views
Help Proving Quotient of Sequences Is Finite
I'm trying prove that $\displaystyle \limsup_{n\to\infty}\frac{a_n}{b_n}$ is finite, when I know that $\displaystyle \frac{a_{n+1}}{a_n}≤\frac{b_{n+2}}{b_n}$.
I'm a little stuck, but have some ...
1
vote
1answer
29 views
Quick question about limits.
Sometimes, when we take limits, especially for roots and ratio tests, we define
lim of sup(a_n). Is this only because when we take the limit of sup we don't have to worry about the existence of the ...
1
vote
2answers
130 views
Probability of limsup
Let $A_1, A_2, A_3, \dots$ be a sequence of independent events on $\left (\Omega, \mathbb A, \mathbb P\right )$ such that $\mathbb P(A_n) < 1$ and $\mathbb P\left ...
3
votes
1answer
63 views
Can we prove $\displaystyle \limsup_{n \to \infty} \sin(n) = 1$?
Can we prove that $\displaystyle \limsup_{n\to \infty} \sin(n) = 1$?
I can prove that the above statement holds assuming that $\displaystyle \frac{\pi}{2}$ is normal (this fact is used somewhat ...
1
vote
1answer
74 views
Prove two inequalities about limit inferior and limit superior
I wish to prove the following two inequalities:
Suppose $X$ is a subset in $\Bbb R$, and functions $f$ and $g$: $X\to \Bbb R$, and $x_{0}\in X$ is a limit point. Then: $$\lim\sup_{x\to ...
6
votes
4answers
177 views
Question in real analysis about liminf and limsup
Can anyone prove this question? I tried but I didn't get any I idea, so I hope someone can solve it.
Let $X_n\leq Y_n$ for each $n\in \Bbb N$. Show that $\liminf X_n \leq \liminf Y_n$ and $\limsup ...
1
vote
1answer
79 views
limit inf /sup if $x_n\leq y_n n$
I have just 2 problems :
1)
Find the $\limsup$ $x_n$ and $\liminf$ $x_n$ where $x_n$$=$ $e^{-n}$.
2)
Let $x_n\leq y_nn$ for every $n\in$$\mathbb{N}$ .
Show that $\liminf x_n\leq\liminf y_n$ and ...
0
votes
2answers
84 views
limsup and liminf are equal when…
I have to prove the following:
If $\forall n\in \mathbb N \colon { \cfrac { 1 }{ 100 } <{ a }_{ n } }$ then $\limsup\limits_{ n\to\infty }\{ \frac { 1 }{ { a }_{ n } } \} =\frac { 1 }{ ...
2
votes
1answer
60 views
Sequence diverges $\iff$ limit inferior diverges
Show that the sequence $X=(x_n)$ in $\mathbb{R}$ diverges to $+\infty$
if and only if $\lim \text{inf}(x_n)=+\infty$. Similarly, show that
$\lim\text{sup}(x_n)=+\infty$ if and only if there is ...
2
votes
1answer
43 views
Is it true that $\liminf ns_n=0$?
Assume $\sum s_n$ be a convergent series and $s_n$ are non negative for all $n$. Is it true that $\liminf ns_n=0$?
Attempts: Intuitively I guess the answer is yes because if the series is convergent, ...
4
votes
2answers
90 views
Inequality between limits inferior
Let $(a_n)$ be a non-zero real sequence with $\left (\frac{a_{n+1}}{a_n} \right)$ bounded. How might we prove that
$$\liminf_{n\to\infty} \, \frac{|a_{n+1}|}{|a_n|} \leq \liminf_{n\to\infty} \, ...
1
vote
3answers
106 views
Limit superior and inferior
How can I find the limit superior/inferior of $a_n$, as $n \rightarrow \infty $?
$$a_n=\frac{n^2+4n-5}{n^2+9}\sin^2\left(\frac{n\pi}{4}\right), n \in \mathbb N$$
I've tried Wolfram|Alpha, but it ...
1
vote
1answer
76 views
$\lim \sup\{X_n\geq x\}$ vs $\{\lim \sup X_n \geq x\}$
Let $(X_n)$ (n is a natural number) be a sequence of real valued random variables.
For any real number $x$, let's define:
$E_x = \limsup \{ X_n \geq x\} $, $F_x = \{\limsup X_n \geq x\} $
If $x$ is ...
0
votes
1answer
20 views
Understanding Proofs Related to Limes and Landau Notation
We had to proove that if $f_1(x) = O(g_1(x))$ and $f_2(x) = O(g_2(x))$ for $g_i(x)$ > 0 then
$i) f_1(x) + f_2(x) = O(g_1(x) + g_2(x))$ and
$ii) f_1(x) + f_2(x) = O(g_1(x)g_2(x))$
Now I have ...
4
votes
1answer
124 views
2
votes
1answer
78 views
Find lim sup $x_n$
Let $x_n = n(\sqrt{n^2+1} - n)\sin\dfrac{n\pi}8$ , $n\in\Bbb{N}$
Find $\limsup x_n$.
Hint: lim sup $x_n = \sup C(x_n)$.
How to make it into a fraction to find the cluster point of $x_n$?
0
votes
1answer
63 views
Sequence of Random Variables
Given
$$\displaystyle \mathbb P(X_n=1)=\frac{1}{n^{\alpha}},~~~\text{and}~~~\displaystyle \mathbb P(X_n=0)=1-\frac{1}{n^{\alpha}}$$ where in $\alpha>0$, $n\geq1$.
How can I prove that ...
0
votes
1answer
60 views
How to prove the limsup equals liminf for a monotone class.
How to prove if a class is monotone, then its limit supremum equals its limit infimum. Example, ${A_{n}}$ is a monotone class with $A_{n} \subset \Omega$, and $A_{1} \subset A_{2} \subset A_{3}... $, ...
1
vote
2answers
193 views
$\limsup c_n\le \max(\limsup a_n,\limsup b_n)$
have a question that im stuck on here
Let $a_n, b_n$ and $c_n$ be three sequences of real numbers.
Suppose $k_n \in [0,1]$ for all $n$. Let $$c_n = (k_n)(a_n) + (1-k_n)b_n\;.$$ Assuming that ...
2
votes
1answer
167 views
lim sup inequality proof - is this the right way to think?
I have tried to read many proofs of this but I'm not sure I get it, so please bare with me.
Show that $\lim_{n \rightarrow \infty} \sup (a_n+b_n) \leq \lim_{n \rightarrow \infty} \sup (a_n)+lim_{n ...
1
vote
1answer
76 views
Proof of limit inequality
Prove that for any sequence $\{x_n\}$ of positive real numbers
$$\lim\text{sup}\sqrt[n]{x_n}\leq \lim\text{sup}\frac{x_{n+1}}{x_n}.$$
My attempt:
Let $A = \lim\text{sup}\frac{x_{n+1}}{x_n}$. ...
1
vote
1answer
76 views
Pointwise convergence, bounded variation, and lim inf's
Suppose that $\{f_n\}_{n = 1}^\infty$ is a sequence of functions in $BV[0, 1]$ that converges pointwise to a function $f$ on $[a, b]$. Show that $V_a^b f \leq \liminf_{n \to \infty} V_a^b f_n$.
I ...
3
votes
1answer
73 views
Is it true that $\limsup \phi\le\limsup a_n?$
Define $\phi={a_1+...+a_n\over n}$ and $(a_n)$ be a sequence of real number. Is it true that $\limsup \phi\le\limsup a_n?$
Intuitively i am wondering why the inequality holds as the sup of a sequence ...
0
votes
2answers
66 views
Product of limsup
Let $f(x)$ be positive and increasing and $g(x)$ satisfy $\limsup_x g(x)=1$.
I want to show $\limsup_x f(x) g(x)=\infty$
Is that true and how do i show it?
I'm thinking that since $f(x)$ is ...
0
votes
2answers
59 views
O-Notation: Limsup vs. Lim
This is the big O-Notation as given today in one of our exercise classes (it is sloppy but just as it was on the blackboard):
$$\begin{align*}
&f = O(g):\quad\limsup_{x \rightarrow a}\, ...
1
vote
0answers
52 views
Probability measures on $\mathbb{T}$ whose Fourier coefficients tend to 1
Let $\mu$ be a probability measure on the complex unit circle $\mathbb{T}$. Are the following two assertions equivalent?
$\limsup_{n\to\infty}|\hat{\mu}(n)|=1$.
There exists an increasing sequence ...
2
votes
3answers
124 views
$\liminf a_n = \sup_{n \in \mathbb{N}}(\inf_{k \ge n}a_k)$
How to prove these equalities? Here $a_n$ is a sequence in $ \mathbb{R} \cup \{- \infty, + \infty \}$
$\liminf_{n\to\infty} a_n = \sup_{n \in \mathbb{N}}(\inf_{k \ge n}a_k)$
and
...
0
votes
3answers
232 views
Two definitions of $\limsup$
Here are two equivalent definitions of $\limsup_{n\rightarrow\infty} a_n$:
Let $u_n=\sup\{a_n, a_{n+1}, a_{n+2},\ldots\}$. Then
$$\limsup_{n\rightarrow\infty} a_n = \lim_{n\rightarrow\infty} u_n =
...
1
vote
2answers
120 views
Are limit superior and limit inferior defined for $z_n$ being a complex sequence?
All the definitions of limit superior and limit inferior I have seen (even in the books about complex analysis) define them for a real sequence only.
What could stop us from defining it as follow for ...
4
votes
1answer
92 views
on the convergence exponent of zeros of entire functions
Let $\{z_j\}$ be the sequence of zeros on an entire function $f$. We define the convergence exponent of $\{z_j\}$ as
$$b=\inf\left\{\lambda>0\ \text{s.t.}\ ...
0
votes
0answers
36 views
order of growth of a counting function
Let $\{a_n\}$ be a sequence of complex numbers. For every $t>0$, define $n(t)=$ the number of $a_n$ satisfying the inequality $|a_n|\leq t$. We call $n(r)$ the counting function for the sequence. ...
-1
votes
1answer
93 views
Compute $\liminf a_k^{1/k}$, $ \limsup a_k^{1/k}$, & $\liminf a_{k+1}/a_k$ and $\limsup a_{k+1}/a_k$ as $k\to\infty$
The sequence
$a_k$ = $$3/2\sum_{k=1}^{\infty}\frac{1}{4^k}$$
Does the series converge? Compute $\liminf a_k^{1/k}$, $\limsup a_k^{1/k}$, $\liminf a_{k+1}/a_k$ and $\limsup a_{k+1}/a_k$ as ...
1
vote
2answers
90 views
Average limit superior
Let $\mathcal{l}_\mathbb{R}^\infty$ be the space of bounded sequences in $\mathbb{R}$. We define a map $p: \mathcal{l}_\mathbb{R}^\infty\to\mathbb{R}$ by
$$p(\underline x)=\limsup_{n\to\infty} ...
1
vote
0answers
42 views
Existence of limit inferior and exterior
First of all, i'm sorry that i don't know what the title should be for this question. Please edit the title if there is a better way to describe this question.
=============
Here's a definition from ...
0
votes
3answers
150 views
What is the definition of limit superior/inferior of real function?
http://en.wikipedia.org/wiki/Upper_limit
Wikipedia has the definition for metric space, but it doesn't have a definition for extended real-field.
It just says, 'Limit inferior and limit superior of ...
0
votes
0answers
53 views
Find the maximum value of $\alpha$ and the minimum value of $\beta$ such that
Find the maximum value of $\alpha$ and the minimum value of $\beta$ such that
$$\biggl(\frac{1}{n}+1\biggr)^{n+\alpha}<e<\biggl(\frac{1}{n}+1\biggr)^{n+\beta}$$
From this I think ...
0
votes
1answer
99 views
$\limsup(a\cdot a_n)=a\cdot \limsup(a_n)$
I know it's quite obvious that
$\limsup(a\cdot a_n)=a\cdot \limsup(a_n)$ for $a$ a real number >0,
but I don't know how to prove it.
My second question is whether the following proof works for:
...
0
votes
3answers
112 views
Fatou's Lemma Strengthened to Equality
I'm trying to use an example to show that Fatou's lemma can not be strengthened to equality. I was given a hint, which I'm not quite sure how to use. I was told that if I look at the one-dimensional ...
1
vote
3answers
193 views
A question on $\liminf$ and $\limsup$
Let us take a sequence of functions $f_n(x)$. Then, when one writes $\sup_n f_n$, I understand what it means: supremum is equal to upper bound of the functions $f_n(x)$ at every $x$. Infimum is ...
4
votes
1answer
93 views
limsup of intersection of events as a subset of intersection of limsups
Let $A_1, A_2, \ldots$ and $B_1, B_2, \ldots$ be two sequences of events in some probability triple $(\Omega, \mathcal{F}, \mathbf{P})$. Now, it is true that $\left(\limsup_n A_n\right) \cap ...
0
votes
2answers
197 views
Limits of infimum and supremum for sequences of functions
I need to show that $-\infty \leq \liminf_{k \to \infty}f_k \leq \limsup_{k \to \infty}f_k \leq \infty$ , where $f_k$ is a sequence of functions from $\mathbb{R}^n$ to $\mathbb{R}$. This seems ...
3
votes
0answers
73 views
Limit superior and limit inferior [duplicate]
Possible Duplicate:
liminf and limsup with characteristic (indicator) function
Suppose $\{E_k\}_{k\geq 1}$ is a sequence of measurable sets. Then we can define ...
