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

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1answer
25 views

Prove $X_{\infty} < \infty$

From Williams' Probability with Martingales: How exactly do we prove $X_{\infty} < \infty$ a.s.? $$E[|X_{\infty}|] = E[|\lim X_n|] = E[|\liminf X_n|] = E[\liminf |X_n|]$$ $$ \le \liminf ...
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0answers
12 views

Clarifications on proof of Doob's Forward Convergence Theorem, warning related to it and proof of a corollary

From Williams' Probability with Martingales: $X_n(\omega)$ does not converge to a limit in $[-\infty,\infty]$ --> Is this supposed to be stronger than $\lim X_n$ does not exist because it's ...
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2answers
37 views

Explicit Integrals and LimInf/LimSup

(a) Show that $f(t):=\int_0^\infty e^{-tx}\frac{sin \space x}{x}dx$ exists for $t>0$ and defines a differentiable function $f$. Calculate $f'(t)$ for $t>0$ and evaluate it explicitly. (b) Prove ...
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0answers
18 views

limsup and liminf of a function explenation

I have a little problem at understanding limsup and liminf of a function at a point y or infinity..I would like to understand the definitions and the theorems formally and intuitively..Can someone ...
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0answers
14 views

helping a limit Calculation problem [on hold]

enter image description hereprevious page no my question enter image description here What I do not understand can't just use 2x^3? why the need to do?
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0answers
16 views

$\liminf{X_n} + \liminf{Y_n} \leq \liminf{(X_n + Y_n)}$ [on hold]

Show that $\liminf{X_n} + \liminf{Y_n} \leq \liminf{X_n + Y_n}$ Then find 2 sequences such that a strict inequality holds: $\liminf{X_n} + \liminf{Y_n} < \liminf{X_n + Y_n}$
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0answers
39 views

Theorem 3.37 in Baby Rudin: $\lim\inf\frac{c_{n+1}}{c_n}\leq\lim\inf\sqrt[n]{c_n}\leq\lim\sup\sqrt[n]{c_n}\leq \lim\sup\frac{c_{n+1}}{c_n}$

Here's Theorem 3.37 in the book Principles of Mathematical Analysis by Walter Rudin, third edition: For any sequence $\{c_n\}$ of positive numbers, $$\lim_{n\to\infty} \inf \frac{c_{n+1}}{c_n} ...
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1answer
27 views

Help solving a Limit quesiton [on hold]

enter image description herewhy can't just use 2x^3? enter image description herePrevious page
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0answers
25 views

Proof $\lim\limits_{n\to\infty} \dfrac{|S_n|}{A_n^{1/2} (\log_2 A_n)^{1/p}(\log_2\log_2 A_n)^{(1+\delta)/p} } = 0$

Let $\{X_k\}$be a random variables sequence and $S_n=\sum_{k=1}^n X_k$. I have $$ \limsup\limits_{n\to\infty} \dfrac{|S_n|}{A_n^{1/2} (\log_2 A_n^2)^{1/p}(\log_2\log_2 A_n^2)^{(1+\delta)/p} } \le ...
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1answer
13 views

$\lim _ {n\rightarrow \infty }\inf \{f_n \leq t \}=\{\lim _ {n\rightarrow \infty }\sup f_n \leq t \}$ Proof and Intuition

For a start how can I read $\lim _ {n\rightarrow \infty }\inf \{f_n \leq t \}$ ? I only seen $\lim _ {n\rightarrow \infty }\inf $ for sets and just don't understand the meaning of $\lim \inf$ for an ...
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1answer
28 views

Proof of the Strenghtened Limit Comparison Test

I'm studying on my own using Bonar and Khoury's Real Infinite Series. I understand the proof of the "regular" Limit Comparison Test( a link to google books, page 23 ) but the book doesn't provide a ...
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2answers
44 views

Theorem 3.19 in Baby Rudin: The upper and lower limits of a majorised sequence cannot exceed those of the majorising one

Here is Theorem 3.19 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition. Let $\{s_n \}$ and $\{t_n \}$ be sequences of real numbers. If $s_n \leq t_n$ for $n \geq N$, ...
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0answers
20 views

Theorem 3.17 in Baby Rudin: Infinite Limits and Upper and Lower Limits of Real Sequences

Here're Definitions 3.15 and 3.16 and Theorem 3.17 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition. Definition 3.15: Let $\{s_n \}$ be a sequence of real numbers ...
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3answers
33 views

Proof about infinite limsups and subsequences

I am working on a final exam study guide, and came across this question: Suppose limsup$(a_n)$ = $\infty$. Prove: There must exist a sub-sequence ${a_n}_k$ such that ${a_n}_k \to \infty$. My initial ...
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1answer
9 views

Equivalence bounded limit superior

Suppose that $(x_n)_{n=1}^\infty$ is a real sequence such that $\limsup_n x_n$ exists. I wish to show that $\limsup_n x_n\le\beta \iff \forall\varepsilon>0\ \ \exists N\ \ \forall n\ge N, x_n ...
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2answers
24 views

Let $A\subset \mathbb{R}$ such that $l=\text{inf }(A)$ exists. Prove that $\forall \epsilon >0 $ there is $a\in A$ in the interval $[l,l+\epsilon)$

I need to prove the following: Let $A\subset \mathbb{R}$ such that $l=\text{inf}(A)$ exists. Prove that $\forall \epsilon >0 $ there is $a\in A$ in the interval $[l,l+\epsilon)$ That's what I ...
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1answer
31 views

Fatou Lemma: Why is $\lim\inf f_n = 0$ where $f_n = \chi_{[n,n+1]}$

In this wildly popular post, there is a claim: I like to remember this by example; specifically let $f_n = \chi_{[n,n+1]}$. Then $\lim \inf f_n = 0$, and $\lim \inf \int f_n = 1$. So $f_n = ...
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1answer
24 views

Convergence of a series implies convergence of another series

Let $a_1,a_2,\cdots$ be a sequence of real numbers with $a_i\geq 0$. If $\sum_{n=1}^{\infty}\frac{1}{1+a_n}<\infty$ then show that $\sum_{n=1}^{\infty}\frac{1}{1+x_na_n}<\infty$ for each real ...
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0answers
34 views

Infimum & Supremum in $\epsilon-\delta$ Proofs

When we are introduced to $\epsilon-\delta$ proofs in a usual first-year introductory course to Calculus, we usually always tend to do one of two things when we attempt to prove limits using the ...
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1answer
12 views

I do not understand proving some limit superior problem.

Prob. Show that $~~~\displaystyle\limsup_{k\to\infty} (a_k+b_k) \le \limsup_{k\to\infty} a_k + \limsup_{k\to\infty} b_k$. Let $A_j=\displaystyle\sup_{k\ge j}a_k,~~~B_j=\sup_{k\ge j}b_k,~~\text{ and} ...
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1answer
52 views

Proof that $\liminf_{x\to\infty}f(x) \leq \limsup_{x\to\infty}f(x)$.

I can't understand this proof from my old lecture notes. $\liminf$ is defined as: \begin{align*} \liminf_{z\to\infty}f(z) = \inf_{x< y} \sup_{y< z}f(z) \end{align*} and $\limsup$ is defined ...
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1answer
50 views

Applying root test to sequence $\frac{1}{2} + \frac{1}{3} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots$

The following is an example from Principles of Mathematics, by Rudin. I've been trying to understand the example but haven't quite grasped it because it seems I can solve it differently. Given the ...
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0answers
19 views

Let $a_n$ be a bounded sequence where n is from 1 to infinite. Prove that lim sup $a_n$ and lim inf $a_n$ exist as n approaches infinite.

Let $a_n$ be a bounded sequence where n is from 1 to infinite. Prove that lim sup $a_n$ and lim inf $a_n$ exist as n approaches infinite. I'm finding difficulty in starting this proof. I have been ...
2
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1answer
20 views

Let ${x_k}={2^{(-1)}}^{k}(1+1/k)$ Find $\liminf x_k$ and $\limsup x_k$.

Let ${x_k}={2^{(-1)}}^{k}(1+1/k)$ Find $\liminf x_k$ and $\limsup x_k$. I tried in this way. First, I split {$x_k$} into two subsequences; $x_{2j}=2+\frac{1}{j}$ and ...
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0answers
51 views

Mathematical proof using sequences

Let $\{x_n\}$ be a bounded sequence. a) Prove that there exists an $s$ such that for any $r > s$ there exists an $M ∈ \Bbb{N}$ such that for all $n ≥ M$ we have $x_n < r$. b) If s is a number as ...
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1answer
29 views

Superior and inferior limits with continuous functions

Given two continuous functions $f$ and $g$, from $\mathbb R$ to $\mathbb R$ and with $f \leqslant g$. If we have a sequence $a_n$ such that it has a limit $a$. Is it true that lim sup $(f(a_n)$, ...
2
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0answers
22 views

Long term behavior of Brownian Motion

Let $(B_t)_{t \geq 0}$ be a Brownian motion. The objective is to prove that \begin{align*} \limsup_{t \to \infty} \frac{B_t}{\sqrt{t}} = \infty. \end{align*} By the scaling property of Brownian ...
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1answer
32 views

Show that $p\{\limsup_{n\to \infty }A_n\geq M\}\geq \limsup_{n\to \infty }p\{A_n\geq A\}$ [closed]

Let $M\in\mathbb R$. How can I show that $$p\{\limsup_{n\to \infty }A_n\geq M\}\geq \limsup_{n\to \infty }p\{A_n\geq M\}\ \ \ ?$$ What I tried is $$p\{\limsup_{n\to\infty }A_n\geq ...
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0answers
197 views

$\liminf\limits_{n\to\infty} \ n \,|\mathopen{}\sin n|$

How would you evaluate the limit inferior of the sequence $n\,|\mathopen{}\sin n|$? That is, $$\liminf\limits_{n\to\infty} \ n \,|\mathopen{}\sin n|$$ Edit. Let $\mu$ be the irrationality measure ...
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0answers
23 views

limsup definition in Alexandroff compactification of C

In the Alexandroff, or one-point compactification of $\mathbb{C}$ (by adding a point $\infty$), consider a function $f:\Omega (\subset \mathbb{C}) \to \mathbb{R}$. I was asked to justify the following ...
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2answers
50 views

example of a convergent series that $\lim \sup |\frac{z_{n+1}}{z_n} | > 1$

Let $(z_n) \subset \mathbb C$, with $z_n \neq 0$. It's known that if $\lim \sup |\frac{z_{n+1}}{z_n} | < 1$, so $\sum |z_n|$ converges, then $\sum z_n$ converges. Can I find an example of a ...
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1answer
27 views

Why isn't there equality in the definition of (upper) semicontinuity?

The "standard" definition of upper semicontinuity at a point $x_0$ in a metric space seems to be $\limsup_{x\to x_{0}} f(x)\le f(x_0)$. However, why is it this weaker condition instead of ...
2
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1answer
109 views

How to prove $\limsup_{n \to \infty} |\sin(n)| = 1$?

Does decimal expansion of $\pi$ contain blocks of zeroes of any integer length? I.e. $0$, $00$, $000$, $\ldots$ I discovered this question, when trying to prove $$\limsup_{n \to \infty} |\sin(n)| = ...
2
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1answer
27 views

The Borel-Cantelli lemma and the rule of product (multiplication principle)

Consider the following random experiment: We play a game where in each round we either win or lose.Let $\{A_n\}$,$n\in\mathbb N$, be a sequence of events where $A_n$ is the event that we win in the ...
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1answer
38 views

$\limsup$ and limits in topological space

I'm trying to generalise a result that holds for metric spaces. Let $(X,\tau)$ be a topological space and $f:X \to \mathbb{R}$. If $x_0 \in X$ is a limit point of $X$, define $$\limsup_{x \to x_0} ...
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1answer
37 views

Definition of $\liminf_{x\to+\infty}f(x)$

Let $f:\mathbb R\to\mathbb R.$ Can someone give me the definition of$$\liminf_{x\to+\infty}f(x)\:\:\text{and}\;\;\limsup_{x\to+\infty}f(x)$$ and a reference in which I can find the properties of ...
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1answer
18 views

An exercise on indicator function in a measure space

Given $(\Omega, \mathbb F, \mu)$ a measure space, $(A_n)$ a sequence of measurable sets, $f: (\Omega, \mathbb F) \to (\mathbb R, \mathbb B(\mathbb R))$ an integrable function such that: $\displaystyle ...
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4answers
30 views

if $\lim \sup a_n = l \in \mathbb R$, then given $\epsilon > 0, \exists N$ such as $a_n < l + \epsilon, \forall n \geq N$

If $\lim \sup a_n = l \in \mathbb R$, then given $\epsilon > 0, \exists N$ such as $a_n < l + \epsilon, \quad \forall n \geq N$. So, Let $(a_{n_k})$ subsequence such as $a_{n_k} \to l$. Given ...
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1answer
83 views

Example that $\lim \sup (x_n\cdot y_n)<\lim \sup (x_n)\cdot \lim \sup (y_n)$

This is a short question, I already managed to prove using definitions that $$\lim \sup (x_n\cdot y_n)\le \lim \sup (x_n)\cdot \lim \sup (y_n)$$ But I'm having trouble coming up with an example such ...
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2answers
42 views

Finding $\liminf$ and $\limsup$ of $A_n = (\sin (n) -1,\sin (n) + 1)$? [closed]

For $n \in \mathbb N^*$, $A_n = (\sin (n) -1,\sin (n) + 1)$. How to show that $\limsup A_n = [-2,2], \liminf A_n = \{0\}$? What if $A_n = (\sin (n -1),\sin (n + 1))$ instead?
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2answers
57 views

How to show that if either $\{a_k\}$ or $\{b_k\}$ converges, equality holds. (lim sup, lim inf) [closed]

I am studying Real analysis by the textbook, Measure and Integral: an introduction to real analysis by Wheeden and Zygmund. This is my first time to study mathematical analysis. However, instructor ...
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3answers
27 views

Prove $\,\displaystyle\limsup_{n\to\infty} s_n=m\,$ if $\,m=\sup\left\{s_n \mid h\ge 1\right\}\lt \infty\,$ and sup not attained

I'm trying to prove the following for my analysis course: Let $\;m=\sup\left\{s_n \mid n\ge 1\right\}\lt \infty\;$ and suppose that the supremum is not attained. Prove that $\,\limsup\limits ...
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0answers
22 views

Proof $\liminf_{n\rightarrow\infty}a_n=\limsup_{n\rightarrow\infty}a_n=\lim_{n\rightarrow\infty}a_n$

I am having some issues with understanding a particular step in a proof given in my book of the theorem that states: "We have that $(a_n)$ is convergent if and only if ...
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4answers
67 views

What does this set theory term mean: $x\in \bigcup_{k=1}^{\infty}\bigcap_{n=k}^{\infty}A_{n}.$?

I have a question for an assignment that involves this term. $$x\in \bigcup_{k=1}^{\infty}\bigcap_{n=k}^{\infty}A_{n}.$$ I have a faint idea of what it means, but perhaps someone can tell me if I am ...
0
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1answer
49 views

Positive, real, bounded sequence $(X_n)$. Prove that $\liminf \sqrt{X_n} = \sqrt{\liminf X_n}$ [duplicate]

Define $(X_n)$ to be a bounded sequence of real numbers such that $∀n∈ℕ, X_n ≥ 0$ Show that $\liminf \sqrt{X_n} = \sqrt{\liminf X_n}$ Side-question: $(X_n) ⊆ ℝ$? How should I go about on this ...
0
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1answer
87 views

Show that $\liminf(\sqrt{x_n})=\sqrt{\liminf x_n}$

Let $(x_n)$ be a bounded sequence of real numbers such that $x_n \ge0$ for each $n \in \mathbb N$. Show that $\liminf(\sqrt{x_n})=\sqrt{\liminf x_n}$. I assume we need to consider the subsequence ...
2
votes
3answers
51 views

Lim inf $ (x_n + y_n)$ less than or equal to lim inf $x_n +$ lim sup $y_n$

I have the question: Show that for bounded sequences $(x_n)$ and $(y_n)$: $$\liminf\limits_{n \rightarrow \infty} (x_n + y_n) \leq \limsup\limits_{n \rightarrow \infty} x_n + \liminf\limits_{n ...
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1answer
46 views

Prove that $\liminf_{n→∞} s_n \le \liminf_{n→∞} σ_n$

I am trying to prove that $$\liminf_{n→∞} s_n \le \liminf_{n→∞} σ_n$$ given that $σ_n=\frac1n(s_1+s_2+\dots+s_n)$. Setting $\alpha = \liminf_{n→∞}s_n$, hence $$\forall \epsilon>0 \ ...
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votes
1answer
49 views

Properties of limsup $\limsup \sqrt{a_n} \le \sqrt{\limsup a_n}$ [closed]

Define $\limsup a_n = \lim_{n \to \infty} \sup \{a_n, a_{n+1},\ldots\}$. Let $(a_n)$ be a positive bounded sequence. Prove or disprove that $\limsup_{n \to \infty} \sqrt{a_n} \le \sqrt{\limsup_{n ...
0
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0answers
20 views

Adherence values of a sequence

Let $\{x_n\}$ be a sequence. If $x_{2n-1}=-\frac{1}{n}$ and $x_{2n}=1+\frac{1}{n}$ and $X_n = \{ x_n, x_{n+1},\ldots\}$, verify that $\inf X_{2n-1} = -\frac{1}{n}$ and $\sup X_{2n-1} - \sup X_{2n} = 1 ...