0
votes
1answer
13 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 ...
1
vote
1answer
44 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
44 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
48 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
54 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
62 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
27 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
13 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
35 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
23 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
26 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
28 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
92 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
42 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
44 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
39 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
48 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
24 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
72 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
42 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
56 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
48 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
33 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
48 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
20 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
58 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
70 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 ...
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
272 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
51 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
104 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
143 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
95 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
77 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
59 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
173 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
86 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
1answer
224 views

Prove that $\limsup a_n = \sup P$ and $\liminf a_n = \inf P$, where $P$ is the set of limit points

Let $\{a_n\}$ be a bounded sequence of real numbers and let $P$ be the set of limit points of $\{a_n\}$. Prove that $\limsup a_n = \sup P$ and $\liminf a_n = \inf P$ my work: Since ${a_n}$ is ...
2
votes
3answers
124 views

Prove: $\liminf_{n \to \infty} s_n \le \limsup_{n \to \infty} s_n$

I am looking over examples and the definitions for this section but I am still not familiar with all the tricks. I appreciate any help with proving this (from hints to maybe a solution. It is only a ...
3
votes
1answer
120 views

Please check if my proof is correct of Monotone Convergent theorem

I was required to prove Monotone Convergent Theorem as a corollary of Fatou lemma,i.e using Fatou lemma to prove the MCT. The hint I was given is let $f_n$ be a sequence of increasing function, ...
1
vote
0answers
58 views

how to do $\limsup$/$\liminf$ of a subset sequence (with an exemple)

My prof give us this exercise: With these definitions: $$E' := \liminf_{k\rightarrow \infty}\; E_k = \bigcup_{k=1}^{\infty} \bigcap_{n=k}^{\infty} E_n$$ $$E'' := \limsup_{k\rightarrow \infty}\;E_k ...
2
votes
2answers
49 views

A doubt on limit supremum

In a book, I see the following : $\limsup_{n \to\infty} |a_n| \geq 1$ implies $\limsup_{n \to \infty} |a_n|^{\frac{1}{n}} \geq 1$. Why ?
2
votes
0answers
227 views

a problem about liminf/ limsup with a continuous function

My Mathematical Analysis III professor gave me this problem: Let $f:(0,1) \rightarrow f((0,1))$ be a continuous function in the standard euclidean metric space $($$\Bbb R$,$d_2$$)$ and let ...
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 ...
0
votes
1answer
131 views

Prove that $\sup \{-x \mid x \in A\} = -\inf\{x\mid x \in A\}$

I need to prove that $\sup \{-x \mid x \in A\} = -\inf\{x \mid x \in A\}$ and am having trouble moving the $-x$ out of the $\sup$ to $\inf$. Another thing is that I don't quite know how to prove $b = ...
2
votes
0answers
40 views

Continuous limsup, Discrete limsup relation

Is it true that for a real function f: $$ \limsup_{t \to \infty} f(t) = \sup_{(t_n)_{n \in \mathbb{N}}, (t_n) \to \infty}\limsup_{t_n \to \infty} f(t_n) $$ ? Where the first limsup is seen in the ...