2
votes
0answers
42 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 $f$?)
0
votes
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
1
vote
1answer
45 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 ...
1
vote
0answers
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 ...
1
vote
2answers
42 views

$\lim \sup$ of a sequence

Let $\{A_n\}$ be a sequence and $\frac{1}{R} = \lim \sup A_n$. Let $\alpha < R$. My question: Why is there $n_0\in \Bbb N $ such that $$A_n < \frac{1}{\alpha}\text{ for any } n\geq n_0$$ Thanks ...
3
votes
0answers
25 views

Order of an entire $f $ is $\limsup_{r \rightarrow + \infty} \frac{\log \log M(r)}{\log r}$ [duplicate]

An entire function is of finite order $\rho$ if $$\rho = \inf \{\lambda \geq 0 \ | \ \exists A, B > 0 \ s.t. \ |f(z)|\leq Ae^{B|z|^{\lambda}} \forall z \in \mathbb{C} \}$$ Write $M(r) = ...
0
votes
1answer
120 views

lim sup of the square equals square of the lim sup

"Suppose that for the sequence of real numbers {an}, lim sup (an) = c > 0 Prove that lim sup (an^2) = c^2" For this question, I tried two ways: 1) Since c is the limsup of {an}, given e >0 and k>0, ...
1
vote
1answer
44 views

A limit involving the exponent of convergence

Let $f$ be an entire non-constant function with at least one zero. If $\{z_{j}\}_{j\in \mathbb{N}}$ are the zeros of $f$, set $$b =\inf\left\{\lambda >0 \ | \sum_{j}\frac{1}{|z_{j}|^{\lambda}}< ...
4
votes
1answer
41 views

Given $c_{i} \rightarrow c$, prove if $c \ge 0$ then $\limsup c_{i} a_{i} = c \limsup a_{i}$

Let $a_{i}$ be a sequence of real numbers and suppose that $\limsup a_{i}$ is finite. Let $c_{i}$ be another sequence and suppose $c_{i}$ converges to $c$. Prove that if $c \ge 0$, then $\limsup ...
2
votes
3answers
142 views

Find a sequence such that $\liminf a_n^{1/n}=1/4,\ \limsup a_n^{1/n}=1/3$

How to construct a sequence $\{a_n\}$ such that $1>a_n>a_{n+1}>0$ for all $n>0$ and $$ \liminf a_n^{1/n}=1/4,\ \limsup a_n^{1/n}=1/3. $$ Thanks.
0
votes
1answer
56 views

Prove that $\sup_k f_k, \inf_k f_k, \lim \sup_k f_k, \lim \inf_k f_k, \lim_k f_k$ (if it exists) are all M-measurable.

If $f_1, f_2, f_3,...$ are $M$-measurable, prove that $\sup_k f_k, \inf_k f_k, \lim \sup_k f_k, \lim \inf_k f_k, \lim_k f_k$ (if it exists) are all M-measurable. My thoughts: We know for any sequence ...
1
vote
3answers
87 views

A question on the upper bound of the limit supremum

My question is: Let $(x_n)$ be a bounded sequence of real numbers. Prove that for every $\epsilon > 0$ and every $N\in\mathbb{N}$ there are $n_1, n_2\geq N$ such that ...
0
votes
1answer
132 views

Find the limit sup and limit inf of a given sequence of sets

Suppose we have a set $X_b = \{\frac{a}{b}:a \in \mathbb{Z^{+}}\} $ where $b \in \mathbb{Z^{+}}$. We want to find $\lim_{b \to +\infty} \inf{X_b}$ and also find find $\lim_{b \to +\infty} \sup{X_b}$. ...
1
vote
2answers
82 views

Trying to understand $\sup$ and $\limsup$ of a sequence.

The following is the sequence and a problem that I am working on. $\{x_n\} = (-1)^n + \frac{1}{n} + 2\sin(\frac{n\pi}{2})$ Find the $\sup$, $\inf$, $\limsup$ and $\liminf$ of this sequence. ...
3
votes
1answer
132 views

Question on $\liminf$ and $\limsup$

Let $f:[0,2\pi]\times \mathbb{R} \rightarrow \mathbb{R}$ a differential function satisfying : $\displaystyle k^2\leq \liminf_{|x|\rightarrow \infty} \frac{f(t,x)}{x}\leq \limsup_{|x|\rightarrow ...
2
votes
1answer
100 views

$\limsup$ of sets $=$ set of $\limsup$

Let $f: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}$ be locally bounded, continuous in the first argument. Consider a converging sequence $\{ x_k \}_{k=1}^{\infty}$, $x_k \rightarrow x$. ...
3
votes
0answers
52 views

Does $\lim_{x\to x_0}\limsup_{y\to y_0}f(x,y)=f(x_0,y_0)$?

I'm wondering about the following situation. Suppose $f\colon\mathbb{R}^2\to \mathbb{R}$ is a function. If $f$ is continuous at $(x_0,y_0)$, why does $$ \lim_{x\to x_0}\limsup_{y\to ...
4
votes
1answer
85 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
49 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, ...
1
vote
3answers
600 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
2answers
356 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 ...
0
votes
2answers
161 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 ...
3
votes
2answers
252 views

Proving that $\limsup a_n$ and $\liminf a_n$ are subsequential limits of $\{a_n\}$

Prove that if $\{a_n\}$ is a sequence, then $\limsup a_n$ and $\liminf a_n$ are subsequential limits of $\{a_n\}$. I don't know the case where $\limsup a_n = \infty$.
5
votes
0answers
679 views

Proof of $\lim\sup(a_nb_n)\leq \lim\sup(a_n)\limsup(b_n)$

Let $a_n>0$ and $b_n\geq 0$, then $\lim\sup(a_nb_n)\leq \lim\sup(a_n)\limsup(b_n)$ My attempt at a proof is as follows. Let $A_n=\sup\{a_n, a_{n+1},...\}$, $B_n=\sup\{b_n, b_{n+1},...\}$, and ...
5
votes
1answer
87 views

Inequality for Fourier transform of measure

I am having trouble with the following question. Let $\mu$ be finite measure on $\mathbb{R}$ and let $\hat{\mu}(\xi) = \int_{-\infty}^\infty e^{-ix \xi} d\mu(x)$ be its Fourier transform. Prove that ...
4
votes
1answer
303 views

Limsups of nets

The limsup on sequences of extended real numbers is usually taken to be either of these two things, which are equivalent: the sup of all subsequential limits. The limit of the sup of the tail ends ...
1
vote
1answer
138 views

Show that $\limsup|s_n|^{1\over n}\le \limsup|{s_{n+1}\over s_n}|$ [duplicate]

Possible Duplicate: Inequality involving $\limsup$ and $\liminf$ limit of $\frac{a_{n+1}}{a_n}$ Show that $\limsup|s_n|^{1\over n}\le \limsup|{s_{n+1}\over s_n}|$ and similarly ...
1
vote
3answers
117 views

A question about suggesting idea to give a formal proof to a theorem about sequence

Theorem Every sequence {$s_n$} has a monotonic subsequence whose limit is equal to $\limsup s_n$. I think to show that there exist a monotonic subsequence is kind of straight forward but I could show ...
1
vote
1answer
393 views

Limit of sequence of sets - Some paradoxical facts

I am particularly confused with alternative formulas describing the inner and outer limits of a sequence of sets in topological spaces. The inner limit of a sequence of sets ...
1
vote
2answers
457 views

$\limsup $ and $\liminf$ of a sequence of subsets relative to a topology

From Wikipedia if $\{A_n\}$ is a sequence of subsets of a topological space $X$, then: $\limsup A_n$, which is also called the outer limit, consists of those elements which are limits of ...
3
votes
2answers
176 views

liminf in terms of the point-to-set distance

Let $\mathcal{X}$ be a normed space and $C\subseteq \mathcal{X}$. We define the point-to-set distance for the set $C$ to be: $$ d_C:\mathcal{X}\ni x \mapsto d_c(x):= \inf_{y\in C}\|x-y\| \in ...
1
vote
2answers
474 views

An exercise on liminf and limsup

Take a function $f:\mathbb{R}\rightarrow(0,+\infty)$ non-decreasing and such that $\mathrm{lim\;inf}_{n\rightarrow+\infty}(f(n+1)-f(n))>0$ then ...
5
votes
1answer
755 views

Inequality involving $\limsup$ and $\liminf$

This may have been asked before, however I was unable to find any duplicate. This comes from pg. 52 of "Mathematical Analysis: An Introduction" by Browder. Problem 14: If $(a_n)$ is a sequence in ...
2
votes
2answers
272 views

Question about $\liminf$ and $\limsup$ of a sequence

Suppose that a sequence $\{x_k\}$ has a clusterpoint (or more..) $c\in\mathbb{R}$. what conclusion, if any, can be drawn about either $\liminf x_k$ or $\limsup x_k$ ? I don't know the conclusion... ...
2
votes
3answers
215 views

$\liminf$ of difference of two sequences

Let ${a_k}$, ${b_k}$ be two sequences in $\mathbb{R}$. Suppose ${a_k}$ converges to $a$ s.t. $\liminf_{k \to \infty} {a_k} = \lim_{k \to \infty} {a_k} = a$. Is it then true that $\liminf_{k \to ...