# Questions tagged [limsup-and-liminf]

For questions concerning the definition and properties of limit superior and limit inferior of sequences of sets or real numbers.

1,256 questions
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### Let $(x_n)$ be a bounded sequence and $u=\limsup x_n$. Let E be set of limits of convergent subsequences of $(x_n)$. How do I prove $u \in E$?

I've been trying to attempt this problem for a long time now. At fist I tried to show that the sequence $(u_n)$, where $u_n = \sup_{i \geq n} x_i$, is a subsequence of $(x_n)$. But this is not true ...
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### Edit: If $\liminf\limits_{n\to\infty}{a_n}>0$ then there exists $a\in \mathbb{R}$ such that $a_n\geq a>0$ for all $n\in \mathbb{N}$

Suppose that the $\beta=\liminf\limits_{n\to\infty}{a_n}>0$ where $\{a_n\}\subset(0,\infty)$ then there exists $a\in \mathbb{R}$ such that $a_n\geq a>0$ for all $n\in \mathbb{N}$. I've always ...
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### $\liminf_\limits{n\to\infty}1_{A_n}(x)=1$ $\implies$ $\lim_\limits{n\to\infty}1_{A_n}(x)=1$?

Source: Partial proof from textbook: I've omitted the case where $x\in A^c$ as it's not relevant. I've also highlighted the part I'm having trouble with in blue. Here is my attempt at explaining ...
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### If a sequence $(u_n)$ be such that its every subsequence has a subsequence that converges to $0$, then $\lim u_n= 0$

Suppose that $(u_n)$ is unbounded above. Then, we pick any arbitrary monotone increasing subsequence $(v_n)$ of $(u_n)$. But by hypothesis, we can find a subsequence of $(v_n)$ that converges to $0$. ...
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### Slightly alternative proof to the converse part of Cauchy's General Principle

I want to prove that: If $\forall \epsilon >0$, $\exists k \in \mathbb{N}$, such that $| u_{n+p}-u_n| <\epsilon$, whenever $n\geq k$, $p\in \mathbb{N}$, then $\{u_n\}$ is convergent. Proof: [...
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### A homework question about partial limits

I could really use some help figuring out this question. The question: ${a_n}$ is a series so that $\lim_{n\to\infty} (a_{n+1} - a_n) = 0$. Prove that its group of partial limits is the closed ...
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### Suppose $x_{n}>0$ and tends to zero.We know its arithmetic mean tends to zero. I wonder which will tend to zero faster.

In detail, $x_{n}>0$ and tends to zero, $S_{n}=\sum_{k=1}^{n}x_{k}$ (so $\lim_{n\to\infty}\frac{Sn}{n}=0$). Let $T_{n}=\frac{nx_{n}}{S_{n}}.$ I think the "average" $S_n/n$ will tend to zero more ...
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### Root test of $\frac{1}{2n}$

Let $a_n=\frac{1}{2n}$ Then $|a_n|^\frac{1}{n}=\frac{1}{2n}^\frac{1}{n}$ And $\frac{1}{2n}^\frac{1}{n}<1$ for every $n\geqslant1$. So $L=\limsup_{n\rightarrow\infty} |a_n|^\frac{1}{n}<1$ By ...
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### Can we conclude $\liminf_{n\to\infty}\operatorname E\left[1\wedge e^{X_n}\right]=0$ from $\liminf_{n\to\infty}X_n=-\infty$?

Let $(X_n)_{n\in\mathbb N}$ be a real-valued process on a probability space $(\Omega,\mathcal A,\operatorname P)$ with $$\liminf_{n\to\infty}X_n=-\infty\;\;\;\text{almost surely}.\tag1$$ Are we able ...
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### How to evaluate $\lim_{N\to\infty}\sup_{x\in(1,\infty)}\big|\sum_{n=N+1}^{\infty}\frac{1}{1+x^n}\big|$?

How to evaluate the following limit $$\lim_{N\to\infty}\sup_{x\in(1,\infty)}\big|\sum_{n=N+1}^{\infty}\frac{1}{1+x^n}\big|\ ?$$ My hunch is that this limit should be zero, but I am unable to prove ...
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### $\liminf$ and $\limsup$ are in $\sigma$-algebra?

Let $F$ be a $\sigma$-algebra of sets of $\Omega$. I have to show that if $(A_n)$ for $n \geq 1$ is a sequence of events such that $A_n \in F$, then $\liminf A_n$ and $\limsup A_n$ are both elements ...
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### Finding the lim inf and lim sup of a sequence

I'm working on the following problem: Find $\lim \sup_n a_n$ and $\lim\inf_n a_n$ of the sequence given by $a_n = 1 + (-1)^n\frac{2n+3}n.$ I've done the following work so far: Let $\{i_n\}_n$ be ...
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### Set sequences $\liminf$ and $\limsup$ - correct?

I'm studying set theory for probability and statistics, and it's important, in order to work with a $\sigma$-algebra, to discuss the concept of $\liminf$ and $\limsup$ for sequences of sets. But in ...
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### What does this notation mean: $\limsup_{\epsilon \to 0} \dots$

In Cohn's measure theory, second edition, p166, there is written: Let $\mu$ be a finite Borel measure on $\mathbb{R}^d$. Then the upper derivate $(\overline{D}\mu)(x)$ of $\mu$ at $x$ is defined by ...
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### If $a_n=n\left(1-\frac{1}{n}\right)^{n[\log n]}$, prove $1\leqslant \liminf a_n$ and $\limsup a_n\leqslant e.$

Prove that: $$\limsup n\left(1-\frac{1}{n}\right)^{n[\log n]} \leqslant e$$ and $$\liminf n\left(1-\frac{1}{n}\right)^{n[\log n]} \geqslant 1.$$ Attempt. Since for $n>2$: 1<[\log n]&...
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### Greatest lower bound of the set : $\{(e^n + 2^n)^\frac1n\ | \; n\in \mathbb{N}\}$

Find the greatest lower bound of the set : $\{(e^n + 2^n)^\frac1n \mid n\in \mathbb{N}\}$ I will find the limit of $a_n := (e^n + 2^n)^\frac1n$ which if exists, must equal the greatest lower ...
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### Convergence of $a_n=\sum\limits_{m=2}^{n} \frac{(-1)^mm}{(\ln(m))^m}$ and $b_n = \sum\limits_{m=2}^{n} \frac{1}{(\ln(m))^m}$

Let $a_1=b_1=0$ and for each $n\geq 2$, let $a_n$ and $b_n$ be real numbers given by $a_n=\sum\limits_{m=2}^{n} \frac{(-1)^mm}{(\ln(m))^m}$ and $b_n = \sum\limits_{m=2}^{n} \frac{1}{(\ln(m))^m}$,...
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### Let $(a_n),(b_n)$ be sequences of positive real numbers such that $na_n<b_n<n^2a_n$

Let $(a_n),(b_n)$ be sequences of positive real numbers such that $na_n<b_n<n^2a_n$ for all $n\geq2$. If the radius of convergence of $\sum a_n x^n$ is $4$, then $\sum b_n x^n$ A)converges ...