3
votes
2answers
42 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 $} ...
0
votes
1answer
51 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 ...
4
votes
1answer
33 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: ...
3
votes
0answers
52 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 ...
1
vote
1answer
52 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
28 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
42 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
87 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 ...
1
vote
1answer
52 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
49 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
50 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
56 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 ...
1
vote
2answers
69 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
28 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
14 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} ...
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
26 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 ...
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
58 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
0answers
32 views

$\limsup \sqrt[n]{(\lambda a_n)} \leq \limsup \sqrt[n]{b_n}$ given pointwise inequality

Let $(a_n),(b_n)$ be non-negative real sequences, $0 < \lambda \in \mathbb{R}$ and $m \in \mathbb{N}$. Further assume $\lambda^\frac{1}{n}a_n^\frac{1}{n} \leq ...
1
vote
2answers
77 views

What does $\liminf_{n\to \infty} f_n$ and $\limsup_{n\to \infty} f_n$ mean?

I have searched the internet, including this website, and I have found some answers, but none which I understand what actually mean. What is $\liminf_{n\to \infty} f_n$? Is it a single value? Is it a ...
1
vote
1answer
107 views

Characterization of lim sup, lim inf

If $(a_n)$ is a real sequence, in lecture we had: $\limsup_{n\to\infty} a_n=a \iff (i)\forall \epsilon >0 \,\exists n_0\in \mathbb{N} :a_n<a+\epsilon$ $ \forall n\ge n_0$ and $(ii) \forall ...
0
votes
1answer
45 views

Product of Limitsuperior of bounded sequences

$\{a_{n}\}$,$\{b_{n}\}$ are two bounded sequences.How can we prove that $\limsup(a_{n}b_{n})$ = $\lim(a_{n})\limsup(b_{n})$. Is it equal to $\lim(a_{n})\lim(b_{n})$? If the sequences are not ...
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 ...
2
votes
1answer
41 views

Limit superior and limit of a sequence

$\{x_n\}$ be a bounded sequence such that for every bounded function $\{y_n\}$, $$\limsup\{x_{n}+y_{n}\} =\limsup\{x_{n}\}+\limsup\{y_{n}\}$$ then prove that {$x_{n}$} is convergent. This is a ...
0
votes
1answer
49 views

relations between the root test and the ratio test

relations between the root test and the ratio test I know the theorem is correct if they are exist $$ \lim\inf\limits_{n\rightarrow \infty} \frac{A_{n+1}}{A_n} \leq \lim\inf\limits_{n\rightarrow ...
0
votes
1answer
25 views

Derivatives from the left and right

I need to give a definition of the derivative from right and from left. (I know it doesn't make a whole lot of sense, but it is supposed to be similar to the same thing as the limits from the right ...
1
vote
1answer
73 views

measure space problem.

Let $(\Omega,\mathcal{F},\mu) $ be a probability space. Let $\delta>0$ and for each $n\in \mathbb{N}$. Let $A_n \in \mathcal{F}$ satisfy $\mu(A_n)\ge\delta$. Prove that the set $A_\infty $ ...
0
votes
1answer
44 views

What are the lim sups and lim infs of a sequence?

What relation, if any, can you state for the lim sups and lim infs of a sequence {an} and one of its subsequences {$a_n$$_k$}?
1
vote
2answers
60 views

$\limsup \sqrt[n]{a_n} \leq \limsup a_n$ if $(a_n)$ is non-negative sequence?

Let $(a_n)$ be a sequence with $0 \leq a_n \leq 1$. Is it possible to show, that $\limsup \sqrt[n]{a_n} \leq \limsup a_n$? Conversely, if $(a_n)$ is a sequence with $0 \leq a_n$ and $\limsup ...
0
votes
1answer
49 views

Help proving a limit exists using limsup and liminf.

Let $X=(x_n)_{n\in N}$ be a bounded sequence of real numbers with $\liminf_{n \to \infty}x_n=\limsup_{n \to \infty}x_n$. Prove that $X$ is convergent. I'm not sure where to start. Any suggestions?
0
votes
1answer
31 views

Oscillation, diameter and limsup

Let $I\subset \mathbb{R}$ be an interval. By definition $\text{diam}f(I)=\sup_{x, x'\in I}|f(x)-f(x')|$, $\text{osc}(f, I)=\sup_{x\in I}f(x)-\inf_{x\in I}f(x)$ and $\text{osc}(f, ...
0
votes
1answer
34 views

$\inf$ of a sequence of $\sup$

let $f_n$ a sequence of right-continuous functions $[0,\infty) \to R$. Assume that for each $T$: \begin{align} & \sup_{\substack{t \in [0,T]\\n \in N}} |f_n(t)|<\infty \end{align} Is it true ...
0
votes
1answer
52 views

$\lim\sup(|s_n|) = 0$ iff $\lim(s_n) = 0$

I need to prove that $\lim\sup(|s_n|) = 0$ if and only if $\lim(s_n) = 0$. I totally get the intuition behind this implication but not sure how to give a formal proof of it. Here is my intuition: for ...
1
vote
0answers
21 views

limsup and quotient of sequences

Let $r\notin\mathbb{Q}$ and $\left\{ a_{n}\right\} _{n\in\mathbb{N}},\left\{ b_{n}\right\} _{n\in\mathbb{N}}$ such that $\lim_{n\to\infty}\frac{a_{n}}{b_{n}}=r$. Find $\limsup a_{n}$ and $\limsup ...
1
vote
1answer
60 views

To prove that $\limsup_{n\to\infty}\frac{1}{a_n} = \frac 1{\liminf_{n\to\infty}a_n}$

I want to prove this equality: $$\limsup_{n\to\infty}\frac{1}{a_n} = \frac{1}{\liminf_{n\to\infty}a_n},\; \forall n\ a_n > 0.$$ What I was thinking about was: Just following the definition, let ...
0
votes
1answer
72 views

$\lim\sup$ vs. $\lim$ as $n \to \infty$

What is the difference between the expressions $$ \lim\sup\left|\frac{a_{n+1}}{a_n}\right| \ \ \ \ \ \mbox{and} \ \ \ \ \ \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right| $$ specifically with ...
0
votes
0answers
117 views

Subsequential limits and lim sup/lim inf

Find the subsequential limits of the following sequences: (Additionally, find the $\limsup s_n$ and $\liminf s_n$) (a) ${[1/2, - 1/2 , 3/4 -3/4, .... (n-1)/n, -(n-1)/n}]$ I have shown the ...
2
votes
0answers
63 views

Proof for $\bigwedge_{n=1}^\infty\bigvee_{k=n}^\infty x_k = \limsup x_n$

How I can show the truth of this relationship? $$\bigwedge_{n=1}^\infty\bigvee_{k=n}^\infty x_k = \limsup x_n$$ The definition of $\limsup x_n$ is $$\limsup x_n = \inf_n\left[\sup_{k\ge ...
2
votes
2answers
273 views

Does $\lim \sup x_{n+1}-x_n=+\infty \implies \lim \dfrac{n}{x_n}=0$

If $(x_n)$ is a real sequence such that $\lim \sup x_{n+1}-x_n=+\infty$ , then must we have $\lim \dfrac{n}{x_n}=0$ ?
1
vote
1answer
53 views

Limit, Limsup clarification needed.

I know that if $(a_n)$ and $(b_n)$ are convergent sequences, and $a_n \leq b_n$, $\forall n\ge N$, then $\lim\, a_n\leq \lim\,b_n$. Now my question is what if we don't know that $(a_n)$ and $(b_n)$ ...
0
votes
1answer
112 views

Limes superior of product of two sequences

I'm an undergraduate mathematics student and I'm trying to understand the basics of limes superior. I recently got stuck on the following question: Let $(a_n)$ be a sequence of numbers and let ...
1
vote
2answers
151 views

Find $\limsup$ and $\liminf$ of a sequence and prove $\lim\ inf \leq \lim\ sup$.

I have a question of finding lim sup and lim inf of $a_n=\frac{1}{n} + (-1)^n$ and prove $\liminf a_n \leq \limsup a_n.$ So the work below is what I did for the first part. $a_{odd\ n} = ...
2
votes
1answer
96 views

Prove $\liminf (s_n\cdot t_n) = \liminf (s_n) \cdot \liminf (t_n)$

I'm having problems with the following proof: If $s_n$ and $t_n$ are sequences, then does $\liminf (s_n\cdot t_n) = \liminf (s_n) \cdot \liminf (t_n)$? Is there a theorem that proves this? Is this ...
0
votes
1answer
82 views

find Lim sup $X_n$ and Lim inf $X_n$?

Question: let $X_{n}=\frac{(n-1)(-1)^{n}}{n}$ find the $\limsup(X_{n})$ and $\liminf(X_{n})$ Can I get someone to help me with this proof? I get that $\frac{(-1)^{n}}{n}$ gives me ...
0
votes
1answer
62 views

lim sup of two sequences

Let $(a_n)_{n \in\ \mathbb{N}}$ a bounded sequence in $\mathbb{R}$. For $n \in \mathbb{N}$ let $$v_n=\sup\{a_k; ~k \geq n\},\quad u_n=\inf\{a_k; ~k \geq n\},\quad s_n=\sup\{|a_k-a_l|; ~k,l \geq ...
1
vote
1answer
187 views

Difference between lim and lim sup?

I know it is a basic question but the definitions in POMA are too rigorous. I need some sort of example to understand what's going on. Can someone please help? Thank you so much
3
votes
0answers
90 views

lim sup intuition for a sequence of sets

lim sup of a sequence of sets $(E_n)$ is defined as $$\bigcap_{n = 1}^{\infty}\bigcup_{k = n}^{\infty} E_k$$ and this means that an element ins lim sup $E_n$ is a member of infinitely many of the ...
1
vote
2answers
387 views

Prove that subsequence converges to limsup

Given a sequence of real numbers, $\{ x_n \}_{n=1}^{\infty}$, let $\alpha =$ limsup$x_n$ and $\beta = $ liminf$x_n$. Prove that there exists a subsequence $\{ x_{n_k}\}$ that converges to $\alpha$ ...