# 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.

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### How to understand that both the lim inf and lim sup of the sets of real numbers is either open or closed?

I am reading the Real Analyais (4th edition) by Royden, H. L., & Fitzpatrick, P. In Section 1.4, it is said that both the lim inf and lim sup of a countable collection of sets of real numbers, ...
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### Prove convergence of $\limsup_{n\to\infty}$

I am new to Real Analysis, and I have found this problem hard to formalize. Problem Let $(p_n)_{n\in\mathbb{N}}$ and $(q_n)_{n\in\mathbb{N}}$ sequences such that $(p_n)\to u$ and $(q_n)\to v$. ...
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### Why does $\lim(\inf (a_n +b_n))=\lim( \inf(a_n) )+ \lim (\inf(b_n))$ fail?

I was working on the proof that for bounded sequences $a_n$ and $b_n$, $\liminf_{n\to\infty}(a_n)+\liminf_{n\to\infty} (b_n) \leq \liminf_{n\to\infty} (a_n+b_n)$. I got to a point where I concluded ...
1 vote
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### Confusion about $\lim\sup$ and its definition as the greatest limit point

I posted a question a few days ago and the most voted answer uses $\lim \sup$, a concept I was not familiar with. I decided to jump ahead and read about $\lim\sup$ to understand the answer, but one ...
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### Equivalent Definitions of Limsup [proof verification]

I've been working on this proof for a few days. While there've been several posts on this, none is not up to my standard for rigor. Below is my attempt, which seems clear to me except for the case ...
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### Prove that $\liminf_i a^{-1}_{i-j}a_i = \liminf_i a_i^{j/i}$ for any sequence $a_i$?

Title pretty much says it all. I saw this claim in some of my lecture notes which says this is a "general fact about sequences". In full: let $(a_i)_{i \in \Bbb{N}} \subset \Bbb{R}_{> 0}$ ...
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### intuition on lim sup and lim inf in probability spaces

So I thought I understand what the lim sup and lim inf are, especially thanks to this post, but now I am presented with a problem from basic probability theory saying: Let $(A_n)$ be a sequence of ...
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My understanding of existence and uniqueness results is that the former condition asserts the cardinality of the "solution set" is positive, whereas the latter implies it is bounded above by ...
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### Construct a sequence $(a_n)_{n=1}^{\infty}$ which has exactly three limit points, at $- \infty, 0,$ and $+ \infty$

Definition. Let us say that a sequence $(a_n)_{n=M}^{\infty}$ of real numbers has $+\infty$ as a limit point iff it has no finite upper bound, and that it has $-\infty$ as a limit point iff it has no ...
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### Existence of a subsequence that has positive limit, given the original sequence is bounded and has a positive limit superior.

Suppose I have a sequence $f: \mathbb N \rightarrow \mathbb R$ such that $0 \le f(n) \le 1$ for all $n$. By the Bolzano Weierstrass theorem, that there must be convergent subsequences of $f$. Suppose ...
1 vote
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### Definition of $\limsup$ from the right or from the left

Let $X$ be a metric space and $E \subset X$. If $f: E \rightarrow \mathbb{R}$, we can define for $a \in E$, the superior limit \begin{align} \limsup_{x \to a} f(x) = \lim_{r \to 0} \left( \sup\{ f(x) ...
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### Questions About Four Definitions of The Upper and Lower Limits of A Sequence

Related questions have been posted here and here. Background I have seen the following four definitions of the upper and lower limits of a sequence from textbooks and MSE posts: Definition 1$\quad$ [...
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### Expressing the event $\{ \limsup_{n \rightarrow \infty} X_n = l \}$

Let $(\Omega, \mathcal{H}, \mathbb{P})$ be a probability space. Let $(X_n)_{n=1}^{\infty}$ be a sequence of real valued random variables s.t. $\forall n \geq 1 \quad X_n$ is $\mathcal{H}$-measurable. ...
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### The limit points of the sequence $x_n=1+\sin\left( \frac{n\pi}{4} \right)$

Consider the sequence \begin{align} x_n= 1 + \sin\left(\frac{n\pi}{4} \right). \end{align} I need to find the limit points of $\lbrace x_n \rbrace$. Notice that \begin{align} x_{8n}&= 1 , \\ x_{8n+...
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### Why is $\{\omega \in \Omega | lim_{n\to\infty}X_n(\omega)=1\}$ in the tail $\sigma$-Algebra?

On a probability space $(\Omega, \mathcal{A}, P)$ I have a series of random variables $X_n, n \geq 1$ and $\mathcal{A}_{\infty}:=\limsup_k \sigma(X_k)$ denotes the corresponding tail $\sigma$-field....
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### Asymptotic Behaviour of $e^{ \alpha \log{( \frac{a}{b})} \log(n)} \times \sum_{t=1}^{n/2} e^ {- \frac{2t}{3} (\log(t) - 3) }$

If $a>b>0$ and $\alpha = \frac{x+1}{2y}$ where $y>0$ and $x\geq 0$. Under which conditions the following summation is asymptotically ($n\to \infty$) upper-bounded by $k n$ where $k$ is a ...
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### What is $\limsup f(x)$ as $x\rightarrow \infty$

Trying to solve Exercise $2.6$ from Real Analysis text by Shakarchi. Integrability of $f$ on $\mathbb{R}$ does not necessarily imply the convergence of $f(x)$ to $0$ as $x \to\infty$. There exists a ...
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### Proving that $\liminf_{n\rightarrow\infty} \log X_n/\log n \le -1$ a.s. where $X_n$ are i.i.d. random variables, uniformly distributed on $[0,1]$

Let $X_n$, $n\in \mathbb{N}$ be independent random variables, uniformly distributed on $[0,1]$. Show that $$\liminf_{n\rightarrow\infty} \frac{\log X_n}{\log n} \le -1 \ \ \text{a.s.}$$ I added my ...
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### How to prove Exercise 8.2.6 from Analysis 1 Terence Tao [duplicate]

I have been stuck a while on the following exercise of Analysis 1 from Terence Tao: Exercise 8.2.6 Let $\sum^\infty_{n=0}a_n$ be a series which is conditionally convergent, but not absolutely ...
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### Is the limsup perserved under monotonically increasing functions?

I've done a bit of digging but I haven't found any sources to confirm that this is true. Take $\limsup{\{x_n\}}=L$ and $f(x): D \to \mathbb{R}$ to be a continuous and monotonically increasing function ...
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### Calculate the limit $\lim_{n \to \infty} \int_1^e x^m e^x (\log x)^n dx$ with limsup and liminf

I have a question about how to prove the limit using $\limsup$. On the other day, I was asked the following problem in the exam: Let $m$ and $n$ be positive integers. Calculate the following limit, ...
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### $\limsup_{n\rightarrow \infty} \frac{x_n}{y_n} = 1$ implies $x_n \geq (1-\epsilon)y_n$

Suppose $x_n$ and $y_n$ are a sequence of real numbers such that $$\limsup_{n\rightarrow \infty} \frac{x_n}{y_n} = 1 \quad \mathrm{and} \quad \liminf_{n\rightarrow \infty} \frac{x_n}{y_n} = -1.$$ ...
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### Proof clarification

The book I'm reading showed a the promise without proving it, after the bolanzo Weirestrass theorem. The theorem is : Let $\left\{{a_n}\right\}^\infty _{n=1}$ , $\left\{{b_n}\right\}^\infty _{n=1}$ be ...
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### Exchanging limit and inferior limit

Let $b(k,M)$ be a real sequence such that for any $k$, $b(k,M)\in [-M,M]$. I know that $\lim_{M\to+\infty} \sup_{k\geq0} |b(k,M)-a(k)|=0$, meaning that $b(k,M)$ converges to $a(k)$ when $M\to+\infty$...
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### Proving inf sup (lim x->y f(xn)) = sup(lim(f(x_n->y))) [closed]

Can someone provide insight into proving this statement: inf sup (lim x->y f(x_n)) = sup(lim(f(x_n-)))
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### Strict inequality

Consider the measure space ($\Omega, \mathcal{A}$), $A \in \mathcal{A}$. Let $f_n: A \rightarrow [-\infty; \infty]$ be a sequence of measurable functions. My script claims \{x \in A: \inf_n \sup_{k \...
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### Constructing the reals using limit of a sequence of sets containing rationals

This is a follow-up to the question: [1] Paradox: Creating an uncountable set of natural numbers. There is also a relation to this q: [2] Limits of sequences of sets. Are the limit points always ...
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