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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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Prove that $a_n$ converges as $a_n$ is a positive sequence, and lim Sup ${a_n}$ lim sup $\frac{1}{a_n}=1$. [duplicate]

Let $a_n$ be a positive sequence. $\underset{n\to\infty}{lim \ sup} \ {a_n} \ \underset{n\to\infty}{lim \ sup} \ \frac{1}{a_n}=1$ Prove that $a_n$ converges. My attempt: First I'll suppose that ...
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About “Principles of Mathematical Analysis” by Walter Rudin Theorem 3.17(a).

I am reading Walter Rudin's "Principles of Mathematical Analysis". There are the following definition and theorem and its proof in this book. Definition 3.16: Let $\{ s_n \}$ be a sequence ...
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29 views

$\limsup$ and integral inequality

I have functions $f$, $g$ :$(0, \infty) \rightarrow (0, \infty)$, and $g$ is invertible and decreasing (I don't know if it is relevant or not in this case). I know also that $\limsup_{x \searrow 0} \...
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Walter Rudin “Principles of Mathematical Analysis” Definition 3.16, Theorem 3.17. I cannot understand.

I am reading Walter Rudin's "Principles of Mathematical Analysis". There are the following definition and theorem and its proof in this book. Rudin didn't prove that $E \neq \emptyset$. Why? ...
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45 views

Stolz-Cesàro $0/0$ case: is $\limsup \frac{a_n}{b_n}\le \limsup\frac{a_{n+1}-a_n}{b_{n+1}-b_n}$?

The general form of Stolz-Cesaro $\infty/\infty$ case states that any two real two sequences $a_n$ and $b_n$, with the latter being monotone and unbounded, satisfy $$\liminf\frac{a_{n+1}-a_n}{b_{n+1}-...
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35 views

Let $f$ be Lebesgueable integrable, then $\lim_{h\to 0}\sup \int_{|x -y| < h}|f(x) - f(y)| = 0$

I must show that Let $f$ be Lebesgueable integrable, then $$\lim_{h\to 0}\sup \int_{|x -y| < h}|f(x) - f(y)|dy = 0$$ for almost every $x \in \mathbb{R}$. Tentative proof: $$|f(x) - f(y)| \geq ...
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1answer
50 views

If $\{a_n\}$ is bounded and non-decreasing, prove that $\liminf b_n = 0$, $b_n = n(a_{n+1} - a_{n})$

Let $\{a_n\}$ be a bounded and non-decreasing sequence of reals, and $b_n = n(a_{n+1} - a_{n})$. (a) Show that $\liminf b_n = 0$ (b) Give an example of a sequence $\{a_n\}$ such that $\{b_n\}$ ...
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1answer
36 views

Intuition behind inequality for measure of $\liminf$ and $\limsup$

For a set $X$ with a $\sigma$-algebra $\xi \subseteq \mathcal{P}(X)$ and $\sigma$-additive $\mu: \xi \rightarrow [0, \infty]$. The following inequality holds for $(A_n)_{n \in \mathbb{N}} \in \xi^\...
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50 views

$\liminf E_{n_0+n} = \liminf\limits E_{n}$ and $\limsup E_{n_0+n} = \limsup E_{n}$ where $E_n$ is a decreasing sequence?

I dropped the $n\rightarrow\infty$ in the title as it was exceeding the character limit. In the book I'm currently reading, the author claims that $\liminf\limits_{n\rightarrow\infty} E_{n_0+n} = \...
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41 views

For a convergent sequence $\limsup b_n = \lim b_n$

Happy new year! I wish to prove the following result to practice with $\limsup: $ For a convergent sequence$ (b_n)$ we know $\limsup b_n = \lim b_n$ If $b_n$ is convergent, the set of ...
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Finding $\limsup_{n\to\infty}\sum_{k=0}^n(-1)^k|\sin k|$

How can we find $$L=\limsup_{n\to\infty}\sum_{k=0}^n(-1)^k|\sin k|,$$ where $|\cdot|$ denotes the absolute value of $\cdot$? According to this answer, we can see the limit does exists. ...
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1answer
51 views

Is $\limsup_{n\rightarrow\infty}\sqrt[n]{|a_n|}=\frac{1}{R}=\limsup_{n\rightarrow\infty}|\frac{a_{n+1}}{a_n}|$?

$R$ is the radius of convergence for a powerseries I will write down my proof but I am not sure whether this is right because I thought $$\limsup_{n\rightarrow\infty}\sqrt[n]{|a_n|}\leq \limsup_{n\...
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33 views

Clarification on step in proof for Ratio test

My thoughts: So apart from this step, the rest of the proof is fairly simple. Now in terms of lim sup, I know that it is just equal to the regular limit for convergent sequences and for bounded, ...
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51 views

Prove any number $c \in [a, b]$ is a subsequential limit if $\lim\inf x_n = a$, $\lim \sup x_n = b$, $a\ne b$, $\lim(x_n -x_{n+1})=0$

I'm trying to solve the following problem: Let $\{x_n\}$ denote a bounded sequence. Prove that any number $c \in [a, b]$ is a subsequential limit of $\{x_n\}$ if: $$ \begin{cases} \lim_{n\to\...
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33 views

Existence of a subsequence converging to limsup

Let $(a_n)$ be a bounded sequence of real numbers, and define $$\beta_n = \sup \{ a_k : k \geq n \}. $$ This sequence converges to a limit, $$\lim_{n \to \infty} \beta_n = \limsup a_n.$$ I'm ...
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1answer
46 views

Proof verification for $\lim \inf x_n + \lim \sup y_n \le \lim \sup(x_n + y_n)$ for bounded $x_n, y_n$

Let $\{x_n\}$ and $\{y_n\}$ denote two bounded sequences. Prove that: $$ \lim \inf x_n + \lim \sup y_n \le \lim \sup(x_n + y_n) \\ $$ We know that both $x_n$ and $y_n$ are bounded hence is their ...
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2answers
35 views

Lim Sup of n-th root of integrable functions

I am working on some analysis problems for a qualifying exam and came across this one which has given me some problems: Let $f_{n}$ be $\textbf{nonnegative}$ measurable functions on a measure space $(...
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sup, limsup, inf, liminf of $x_k=\frac{1}{k}+\cos(\frac{k\pi}{2})$ [check] [closed]

I am considering the sequence $x_k=\frac{1}{k}+\cos(\frac{k\pi}{2})$ for $k\in\mathbb{N}$ for $\sup\{x_k\mid k\in\mathbb{N}\}$ I got $\frac{5}{4}$ (k=4) $\limsup_{n\rightarrow\infty}$ I got $1 $ $\...
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46 views

Show that $\limsup_{k \to \infty} 2^{-k} N_k = 0$ where $N_k$ is the number of $a_n \geq 2^{-k}$.

Let $\{a_n\}_{n \geq 1}$ be a non-negative sequence of reals such that $\sum_{n \geq 1} a_n$ converges to $s$. Define $N_k = |\{n \in \mathbb{N} : a_n \geq 2^{-k}\}|$. Show that \begin{equation} \...
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1answer
32 views

Does $\liminf$ distribute?

I know the property that for sets $A_n,B_n$ $$ \limsup_{n\to\infty}(A_n\cup B_n)=\limsup_{n\to\infty}(A_n)\cup \limsup_{n\to\infty}(B_n) $$ holds. I'm curious if it also holds that $$ \liminf_{n\to\...
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Solution verification: evaluate $\lim\limits_{n \to \infty}\frac{1^{\lambda n}+2^{\lambda n}+\cdots+n^{\lambda n}}{n^{\lambda n}}$ where $\lambda>0.$

Problem Evaluate $$\lim_{n \to \infty}\frac{1^{\lambda n}+2^{\lambda n}+\cdots+n^{\lambda n}}{n^{\lambda n}}$$ where $\lambda>0.$ Solution Denote $$S_n:=\sum_{k=1}^{n}\left(\frac{k}{n}\right)^{\...
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1answer
54 views

How can $\limsup_{x \to x_0} f(x) = f(x_0)$ for $f$ discontinuous at $x_0$?

My textbook says $$\limsup_{x \to x_0} f(x) = \max\{f(x_0), \lim_{h\to 0^+} f( x_0 + h), \lim_{h\to 0^-} f( x_0 + h)\}$$ Assuming $f(x_0)$ is distinct from the latter two values, how can $\limsup f(...
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Prove limit and liminf of series

I am self-studying convergence of sequence and series and have some hard time doing the following problem: Suppose $\sum_{k=0}^{\infty} a_kb_k^2 < \infty$. (a) If $\sum_{k=0}^{\infty}a_k = \infty$ ...
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1answer
29 views

Limes superior of $X_n>\alpha \ln(n)$

Given the random variable $X_n$, to be exponential distributed with parameter $1$, we define $$A_n:=\left\{X_n>\alpha \ln(n)\right\}.$$ We've shown in the classes that for $\alpha \leq 1$, $A_n$ ...
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1answer
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Theorem 3.37 in Baby Rudin: Any Counter-Examples In The Other Case?

Here is Theorem 3.37 in the book Principles Of Mathematical Analysis by Walter Rudin, 3rd edition: For any sequence $\left\{ c_n \right\}$ of positive numbers, $$ \lim\inf_{n\to\infty} \frac{c_{...
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1answer
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Compute liminf, limsup

I want to compute liminf and limsup of $\left( (-1)^{n^3} \left( 1+\frac{1}{n}\right)^n\right)$. I have thought the following so far: From definition we have that $\lim \inf x_n=\lim_{n \to \infty} \...
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Evaluating $\liminf_{n\to\infty}n\{n\sqrt2\}$

How can we evaluate $$\liminf_{n\to\infty}n\{n\sqrt2\},$$where $\{\cdot\}$ denotes the fractional part of $\cdot$? The first thing came to my mind is Pell's equation $x^2-2y^2=1$. Knowing that $\...
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1answer
34 views

Equivalence of limit supremum of sequence of sets and sequence of functions

Let $X_1,X_2,\dots$ be a sequence of real-valued random variables where $X_n:\Omega \to \mathbb{R}$. For any sequence of events $A_n \subset \Omega$ define $\limsup_nA_n := \bigcap_{n=1}^{\infty}\...
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1answer
35 views

If $\lim(a_{n+1} - a_n) = 0$, prove that $P(a_n) =[\lim\inf(a_n),\lim\sup(a_n)]$

If $\{a_n\}$ is a series, and $\lim(a_{n+1} - a_n) = 0$, prove that $P(a_n) = [\lim\inf(a_n),\lim\sup(a_n)]$ Basically, I wish to prove that the set of all partial limits of the series $a_n$ (as $n\...
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1answer
39 views

find $\limsup$ and $\liminf$ of $a_n=\{\sqrt{n} - \lfloor\sqrt{n}\rfloor$ $n\in\mathbb{N}\}$ [duplicate]

I have to find $\limsup$ and $\liminf$ of $a_n=\{\sqrt{n} - \lfloor\sqrt{n}\rfloor : n \in\mathbb{N}\}$ I suppose that I have to relate to subsequences . First one is $a_{n_k}$ for all n that ...
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1answer
46 views

Proving that continuity and the lim sup of a given set being 0 are equivalent

Suppose we have a function defined as follows: $\alpha(f,x_o)=\limsup\{|f(a)-f(b)|:a,b\in (x_o-\frac{1}{n},x_o+\frac{1}{n})\}$ with $f:R\rightarrow R$ and $x_o \in R$. I need to prove that $\alpha=0$...
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for bounded series $x_n$, prove that $\limsup(1/x_n) = 1/ \liminf( x_n)$

Stuck on a proof about series: i know that $ 0 < a \leq x_n \leq b, $ $a$ and $b$ are real numbers (not infinite) and from this it follows that $1/x_n$ is also bounded by $1/b$ and $1/a,$ which ...
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Why is $\inf g \sup g = \frac{9}{16} $?

Consider this question here : Why is $\sup f_- (n) \inf f_+ (m) = \frac{5}{4} $? Call that conjecture about $\frac{5}{4} $ conjecture $1$. Let $g(n) = \prod_{i=0}^n (\sin^2(n) + \frac{9}{16}) ) $ ...
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1answer
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Prove that the liminf of a sequence is equal to its smallest subsequential limit

$\ a:= \liminf_{n\to \infty}$ $a_n$ = $lim_{n\to \infty}$ $inf_{k\geq n}$ $a_k$ $a$ $\in$ $R$ I am trying to show this by using cases and assuming the equality doesn't hold- to come to a ...
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1answer
38 views

Divergence of limit of function

Let $f$ be a bounded real-valued function on a subset of $\mathbb{R}$ and let $x_{0} \in \mathbb{R}$ be a cluster point with respect to $A$. Suppose that $\lim_{x\to x_0} f(x)$ does not exist. Show ...
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1answer
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Sequence $(f_n) \to f_n$ on $S \subseteq \mathbb{R}$ converges uniformly iff $\lim_{n \to \infty} \sup \{f(x)-f_n(x)| : x\in S\} = 0$

I am having trouble proving the converse. Namely, if $$\lim_{n \to \infty} \sup \{|f(x) - f_n(x)| : x \in S \subseteq \mathbb{R}\} = 0$$ then the sequence $(f_n)$ converges uniformly to $f$ Any ...
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A limit involving exp and log

Consider two sequences $\{a_n\}_{n=1}^\infty$ and $\{b_n\}_{n=1}^\infty$ where $a_n\geq 0 \forall\ n$ and $b_n\geq 0\ \forall n$. Is it true that $$\limsup_{n\rightarrow\infty}\frac{1}{n} \log\left(e^{...
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1answer
46 views

Suppose $F$ is continuous on $[a,b]$.Show that $D^+F(x)=\limsup_{h\to 0\,h\gt0} \frac{F(x+h)-F(x)}{h} $is measurable.

Suppose $F$ is continuous on $[a,b]$.Show that $$D^+F(x)=\limsup_{h\to 0\,h\gt0} \frac{F(x+h)-F(x)}{h} $$is measurable. [Hint:the continuity of F allows one to restrict to countably many h in ...
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Example of $\lim\inf x_n+\lim \inf y_n<\lim \inf(x_n+y_n)$

I am looking for an example demonstrating that $\lim\inf x_n+\lim \inf y_n<\lim \inf(x_n+y_n)$ but for the life of me i can't find one. any suggestions?
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Lim inf of a sequence of sets and the convergence of the its series of the indicator function

(En) sequence of subsets of X. If x belongs to the lim inf(En) How i can prove that the series over N of 1(En)(x) converges? 1(En) : the indicator function
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Proving $x_k-\left(\limsup_{n\to\infty} x_n\right)<\epsilon,\forall k\ge M$ for some $M\in\mathbf{N}$

Is the following argument correct? Suppose $\{x_n\}$ is bounded sequence, and $\epsilon>0$ is given. Prove that there exists a $M$ such that for all $k\ge M$ we have $$ x_k-\Bigl(\limsup_{n\to\...
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0answers
12 views

Implications of $a_l\not\in\{x_k:k\ge l\}$ where $a_n:=\sup\{x_k:k\ge n\},\forall n\in\mathbf{N}$.

Is the following argument correct? Propostition. Suppose $\{x_n\}$ is bounded sequence, $a_n:=\sup\{x_k:k\ge n\}$ as before. Suppose that for some $l\in\mathbf{N}$, $a_l\not\in\{x_k:k\ge l\}$. Then ...
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2answers
46 views

A question about limsup of the sequence of the average

Let $\{a_n\}$ be a sequence of real non-negative numbers Define $S_n = \frac{\sum_{i=1}^n a_i}{n}$ Prove that $$\liminf(a_n) \leq \liminf(S_n) \leq \limsup(S_n) \leq \limsup(a_n)$$ I wanted to ...
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2answers
77 views

limsup equality proof

Let $(\Omega, \mathscr{F},P)$ be a probability space and $\{A_n\}_{n\geq 1}$ a sequence in $\mathscr{F}$. Prove that $\limsup (A_n) \cap \limsup (A^{c}_{n}) = \limsup (A_n \cap A^{c}_{n+1})$. Since ...
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1answer
33 views

Lim inf of addition of sequences

For an exercise, I am trying to prove that if $\liminf (x_n)=\liminf (y_n)=1$, and $\lim(x_n+y_n)=2$, then $\lim(x_n)=\lim(y_n)=1$. $x_n$ and $y_n$ are both bounded sequences as well. I can see why ...
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1answer
17 views

Moment generating function is finite given limsup property

For a random variable $X$ define the moment-generating function $M_X(t) = \mathbb{E}[e^{tX}]$. If it is known that $$\limsup_{x\to\infty}\frac{\log\mathbb{P}(X>x)}{x} =- c < 0,$$ how can it be ...
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1answer
27 views

Show $x_n \in \left[\liminf(x_n) - K, \limsup(x_n) + K\right] $ for all $n \geq N$ (some natural number N)

Given a bounded sequence of real numbers I want to show $x_n \in [\liminf(x_n) - K, \limsup(x_n) + K]$ for all $n\geq N,$ where $N$ is some natural number, and for all $K>0.$ Attempt at a ...
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3answers
53 views

Prove $\liminf$ definition

This is homework assignment. Let $(x_n)$ be bounded sequence. Prove following equation $$\liminf_{n \rightarrow \infty}\, x_n = \max \{ B \in \mathbb{R} : \forall \varepsilon > 0 \{n \in \...
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1answer
76 views

Is $\liminf \limsup$ always greater than or equal to $\limsup \liminf$?

Lets say we have a function $f:X \times Y \to \mathbb R$. Is it always true that $$ \liminf_{y \to b} \limsup_{x \to a} f(x,y) \geq \limsup_{x \to a} \liminf_{y \to b} f(x,y)$$ ? This question was ...
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2answers
94 views

Mean of two numbers by infinite sequences

Consider two numbers $a$ and $b$, and the following sequence alternating between even and odd positions: $$ a+2b+3a+4b+5a+6b\ldots, $$ If we ''normalize'' $$ \frac{a+2b+3a+4b+\ldots}{1+2+3+4+\ldots}, $...