Tagged Questions

4
votes
1answer
61 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
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, ...
1
vote
3answers
102 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
184 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
63 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 ...
1
vote
2answers
150 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
279 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
63 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
199 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
116 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
106 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
282 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
346 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
165 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
346 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
452 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
235 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
161 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 ...