For questions concerning the definition and properties of limit superior and limit inferior.

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48 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 ...
2
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2answers
41 views

Sandwiching Limsups & liminfs of expectations

Why is it that if we sandwich a liminf of an expectation between two equal quantities we get that the limit exists? Can we somehow deduce the limsup from that and conclude that it's the same or am I ...
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1answer
29 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: ...
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1answer
17 views

Coin tossing, two heads always followed by two tails - lim sup necessary?

In Bernoulli Space $\Omega$, let $E_n$ be the event that the $n$th toss is heads. Write down a formula in terms of the $E_n$ for the following event: “Every time two Heads appear in succession, the ...
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0answers
50 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 ...
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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 $f$?)
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1answer
47 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, ...
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1answer
27 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 ...
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1answer
41 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 ...
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2answers
83 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 ...
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123 views

A limsup version of the Principle of Uniform Boundedness?

Suppose $X$ is a Banach space and let $\{f_\alpha\}$ be a net of continuous linear functionals satisfying $\limsup_{\alpha} | f_\alpha(x) | < \infty$ for each fixed $x \in X$. Is it true that ...
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1answer
51 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 ...
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1answer
46 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 ...
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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 ...
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1answer
37 views

IF $\sum_{n=1}^{\infty}P(|X_n|>c)=\infty\forall c>0$, then $\limsup_{n\rightarrow\infty}|S_n|=\infty$ a.s.

If $S_n=\sum_{i=1}^n X_i$ where $X_i,i\ge 1$ are iid and $\sum_{n=1}^{\infty}P(|X_n|>c)=\infty\forall c>0$, then $\limsup_{n\rightarrow\infty}|S_n|=\infty$ a.s. I tried this but got stuck: Let ...
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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 ...
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1answer
45 views

Trying to prove lim inf $(A_n) \subseteq$ lim sup $(A_n)$

I am trying to show that for a sequence of sets $(A_n)$ $$ lim \ inf \ (A_n) \subseteq lim \ sup \ (A_n)$$ I can see why this is true intuitively as follows - $x \in lim \ inf \ (A_n)$ $\implies ...
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1answer
30 views

Applying limit inferior

In the solution to this question while attempting to prove by contradiction it states that if we assume $|x(t)|\nrightarrow \infty $ then $M:=$lim inf $_{t\uparrow b}|x(t)|$ Why can I just not use ...
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90 views

Proving $\limsup\frac 1 {a_n}=\frac 1 {\liminf a_n}$ and $\limsup a_n\cdot \limsup \frac 1 {a_n} \ge 1$

Let $a_n$ be a sequence such that $\forall n\in \mathbb n: 0<a\le a_n\le b <\infty.$ Prove: $\displaystyle\limsup_{n\to\infty}\frac 1 {a_n}=\frac 1 ...
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65 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 ...
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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 ...
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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 ...
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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} ...
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2answers
59 views

When can I interchange $\liminf$ and integral?

Let $g_n:[0,T] \to \mathbb{R}$ be a sequence. When can I say that $$\int_0^T \liminf_{n \to \infty} g_n(t) \leq \liminf_{n \to \infty}\int_0^T g_n(t)?$$ Under what circumstances?
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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 $. ...
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1answer
24 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 ...
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2answers
328 views

$\int_0^1f(x)dx=1, \int_0^1xf(x)dx=\frac16$ minimum value of $\int_0^1f^2(x) dx$?

Let $f(x)\geq0$ be a Riemann integrable function, and $$\int_0^1f(x)\,\mathrm dx=1, \int_0^1xf(x)\,\mathrm dx=\frac16.$$ Find the minimum value of $\int_0^1f^2(x)\,\mathrm dx$ Cauchy-Schwarz ...
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1answer
67 views

A couple of inequalities (explanation needed), how to show $\liminf_{k \to 0}k^{-1}\int(\nabla u - \nabla v)\nabla (T_k(b(u)-b(v))) \geq 0$

Let $T_k(x) = \max\{-k, \min(x,k)\}$, a truncation function at levels $k$ and $-k$. Let $b$ be a Lipschitz increasing function with $b(0)=0$ and $b'$ and $b^{-1}$ also Lipschitz. I have seen it ...
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1answer
35 views

Prove $\{Z=0\}\subset\limsup\limits_{n}\{X_n<\epsilon\}$

Let $(X_n)_{n\in\mathbb N}$ be independent real random variables, with values in $(0,\infty)$. Consider the random variable $Z(w):=\inf\limits_{n\in\mathbb N}X_n(w)$. Prove that for every fixed ...
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1answer
72 views

Show that $\limsup_{n\rightarrow\infty} \|x_n\|^{\frac{1}{n}}=1$

Let $(x_n)$ be weakly convergent, but not norm convergent, sequence from a Banach space. Show that $\limsup_{n\rightarrow\infty} \|x_n\|^{\frac{1}{n}}=1$. Any help?
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1answer
26 views

Interchaging $P(\mathrm{limsup})$ with $P(\mathrm{limit})$ for $P$ a probability measure.

I have been going through Resnick's 'A Probability Path', and at one point he is trying to prove a version of Fatou's lemma: $$P(\liminf_{n\rightarrow\infty}A_n)\le\liminf_{n\rightarrow\infty}P(A_n)$$ ...
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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
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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
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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 $\lim\inf_{x\to0}\frac{xf''(x)}{f'(x)}\geq1$. ...
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1answer
30 views

updated: lim inf, lim sup and bertrand's postulate for a function

Define $q:\mathbb{N}\rightarrow\mathbb{Q}^+$ with, $q(n)=min\{\alpha\in \mathbb{Q}^+\mid \exists \;p\in \mathbb{N},\; p \text{ prime and } p \in (n,\alpha n]\}$. Is the ...
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1answer
31 views

Complex analysis problem, using limsup

Im trying to solve the following problem. Let $\Omega \in \mathbb{C}$ an open bounded set, let $f\colon \Omega \to \mathbb{C}$ be holomorphic, and supose there exists a constant $M \geq 0$ wich ...
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31 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 ...
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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 ...
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1answer
98 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 ...
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0answers
25 views

Determining the expression of supremum correctly - $\sup_{x\in [0,1]} |f-f_{n}|$.

Let $f_{n}$ be a sequence of functions and let $f=\lim_{n\to\infty}f_{n}$. If $f_{n}=nx^{n}(1-x)$ then $f=0$. How do I determine the following expression $$ \sup_{x\in [0,1]} |f-f_{n}|=\sup_{x\in ...
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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 ...
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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
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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 ...
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2answers
33 views

Counterexample for limsup

Statement: $\limsup c_n a_n = c \limsup a_n$ Please help find a counterexample to this statement if $c<0$. Edit: also suppose $c_n$ -> c and $\limsup a_n$ is finite
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2answers
39 views

Prove that limsup is the supremum of the limit points

Let ${a_n}$ be a bounded sequence of real numbers, and let P be the set of limit points. Prove that $\limsup a_n = \sup P$. I have proved that there must be a subsequence that converges to ...
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1answer
20 views

Is this limsup calculation correct?

$a_n=1$ if $n=2^k$ for $k>0$ $a_n=\frac{1}{n!}$ otherwise a) Find limsup $\displaystyle \frac{|a_{n+1}|}{|a_n|}$ - I think this is infinity because we can find a term that is 1/something!, but ...
0
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1answer
26 views

Product of lim sups

Suppose lim sup $a_n$ is finite, and $c_n -> c$ Prove that if $c \geq 0$ lim sup $a_n c_n$ = c lim sum $a_n$ and find a counterexample to this if $c <0$. Is there a rule that the product of ...
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1answer
34 views

limsup of series

Find limsup $|a_j|^{1/j}$ $\displaystyle a_j = \sum_{j=1}^\infty \frac{(1+1/j)^{2j}}{e^j}$ Since the limit of the numerator is $e^2$, is it correct that the limsup is equal to 0? How can I write out ...
0
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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 ...
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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 ...