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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2answers
33 views

Why does every sequence of real numbers have an upper limit?

Let $\{a_n\}:\mathbb{N}\to\mathbb{R}$, and we define $E$ to be the set of all sub-sequential limits of $\{a_n\}$ as well as possibly the symbols $\infty$ and $-\infty$ if there are some sub-sequences ...
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43 views

Let $(a_d), (b_d)$ be real nets satisfying $\limsup a_d = \lim b_d = ∞$. Is there $f:ℝ\to ℝ$ such that $\limsup \frac{a_d}{f(b_d)}=\lim f(b_d) = ∞$?

Let $(D, \geq)$ be a directed set, and let $(a_d)_{d\in D}$, $(b_d)_{d\in D}$ be real-valued nets satisfying $$\limsup_{d\in D} a_d = +\infty \quad \text{and} \quad \lim_{d\in D} b_d = +\infty.$$ (...
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31 views

Show that $a = \sup \{ a_n | n \in \mathbb{N} \} = \limsup\limits_{n \to \infty} a_n$

Let $a \in \mathbb{R}$ be an accumulation value of the real sequence $(a_n)_{n \in \mathbb{N}}$ and an upper bound of the set $\{a_n | n \in \mathbb{N}\}$. Show that $$a=\sup\{ a_n | n \in \mathbb{N} \...
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Construct a sequence of functions in $L^p(\mathbb{R})$ such that $f_n$ converges to zero in $L^p$. [closed]

I'm trying to find a sequence of functions $f_n\in L^p(\mathbb{R})$, with $1\leq p < \infty$ such that $\lVert f_n \rVert_{L^p}\to 0$. Moreover for all $x\in \mathbb{R}$ the sequence satisfy two ...
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1answer
23 views

Existence of subsequences $a_{n_k}$ that converges to a value between $\liminf a_n$ and $\limsup a_n$

I am stuck on this question: Suppose $a_n$ is a sequence with $\liminf a_n=s$ and $\limsup a_n=t$ where $s$ and $t$ are real number. Then there must exist some subsequence $a_{n_k}$ that converges to $...
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1answer
34 views

If lim inf $a_n$ = $\infty$ and lim sup $a_n$ = $-\infty$, $a_n$ diverges?

I know that a sequence is a convergent sequence if and only if it is a Cauchy sequence so lim inf $a_n$ = $\lim sup a_n$. Then, if $\liminf a_n = \infty$, then can we conclude $\lim a_n=\infty$ ? What ...
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1answer
33 views

How to “explain in words” this expression involving liminf

Let $\Omega\subset\mathbb{R}^N$ and $F:\Omega\times\mathbb{R}\to\mathbb{R}$ be a function and consider the expression $$ \liminf_{| u|\to+\infty} \frac{F(x, u(x))}{|u|^p +| u|^q} <0,$$ for some $p, ...
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1answer
40 views

Why is lim sup necessary

I have hard time understanding why the concept of lim superior is necessary, for example the ratio test. There are two ways, it seems of phrasing this: (Abbott, Understanding Analysis p.78) Given $\...
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1answer
31 views

Comparing two events

Let $X_n$ be a sequence of real random variables, $X$ a real random variable and $\varepsilon > 0$ Let $A_n$ be the event $ \{ | X_n - X | \geq \varepsilon \} $ and $B_n$ the event $ \{ \sup_{k \...
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How to prove $\limsup_{n\to \infty} \sqrt[n]{|a_n|}=\limsup_{n\to \infty} \sqrt[n]{|a_{n+1}-a_n|}$ if $\{a_n\}$ diverges [duplicate]

$\{a_n\}$ is a diverging sequence. I have proven that $\limsup_{n\to \infty} \sqrt[n]{|a_{n+1}-a_n|}\geq1$ by contrapositive. Then, we need to prove $\limsup_{n\to \infty} \sqrt[n]{|a_n|}=\limsup_{n\...
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Alternative proof of generalized Borel-Cantelli without the use of the finite increment martingale limit property

I'm working through a proof outline of the generalized Borel-Cantelli lemmas. Following the outline, I've shown that for $A_k\in\mathcal F_k$ and $$X_n=\sum_{k=1}^n 1_{A_k}, Y_n=\sum_{k=1}^n P(A_k\...
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1answer
30 views

Finding all cluster points of a sequence and proving it rigorously

This is a problem from Kenneth Brown's Intro to Real Analysis: Find all the cluster points of the sequence: $$x_n = \cos\left(\frac{n\pi}{4}\right) + \sin\left(\frac{n\pi}{2}\right) + 2^{-n} + (-1)^n$$...
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1answer
57 views

Compute $\lim_{n\to \infty} \sup A_{n}$ and $\lim_{n\to \infty} \inf A_{n}$, where $A_{n} = [0,a_{n}]$

I want to compute the limit (if exists) of the sequence of sets defined by $A_{n} = [0,a_{n}]$. Where $\{a_{n}\}$ is a sequence of nonnegative numbers that converges to $a\geq 0$. I know that I have ...
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1answer
38 views

Find limit definitions manually (without WolframAlpha)

There was a question where $\quad0\le \big(3-\sin(2\pi x)\big)\sin(\pi x - \frac{\pi}{4})- \sin(3\pi x +\frac{\pi}{4})\le 2\sqrt{2}\quad $ and the user wanted to know what values of $x$ would be ...
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46 views

Why did they take the limsup and not the limit

$"$Let $f:\mathbb R^n\rightarrow\mathbb R$ be differentiable... The directional derivative of $f$ at the point $x$ in the direction $h$ is given by $Df(x,h)=\lim_{t\rightarrow 0}\frac 1 t(f(x+th)-f(x))...
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1answer
43 views

Proof of $\varliminf a_n + \varlimsup b_n \leq \varlimsup (a_n + b_n) $ using the subsequential definition

$a_n$ and $b_n$ are assumed to bounded sequences. The definition I'm strictly adhering to is $\varliminf a_n = \inf\{\text{subsequential limits of } a_ n \} $ $\varlimsup a_n = \sup\{\text{...
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1answer
54 views

Non-convergence of a sequence implies 'consecutive' partial limits

Let $n,d$ be fixed positive integers, and let $X^k=(x_1^k, \dots, x_n^k)$ be a bounded sequence, where $x_i^k \in \mathbb{R}^d$. (Every element of $X^k$ is an $n$-tuple of points in $\mathbb{R}^d$). ...
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4answers
36 views

Find the radius of convergence of $\sum_{n=1}^\infty (\frac{5n^3+2n^2}{3n^3+2n})z^n$

Radius of convergence of $\sum_{n=1}^\infty (\frac{5n^3+2n^2}{3n^3+2n})z^n$ I know Hadamards lemma says that $\frac{1}{R}=\lim\sup \vert a_n\vert^{1/n}$ But I'm not sure how to compute $\lim\sup \...
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1answer
25 views

Prove $\sup\{r\geq 0: \exists M\in\mathbb{R}[\forall n\in \mathbb{N} [\, |r^n a_n|<M\ ]] \}= \sup \{ r\geq 0 :\sum_n |r^n a_n| \in \mathbb{R} \}$

Given a complex power series $\sum_n a_n z^n$ for $z,a_n\in\mathbb{C}$, let $$A:=\{r\geq 0: \exists M\in\mathbb{R}[\forall n\in \mathbb{N} [\, |r^n a_n|<M\ ]] \}$$ $$B := \{ r\geq 0 :\sum_n |r^n ...
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$\Vert x_{n +1}-x^*\Vert\leq c\Vert x_n-x^*\Vert ^p\Rightarrow p=\liminf\frac{\ln\Vert x_{n+1}-x^*\Vert}{\ln\Vert x_n-x^*\Vert}$

$x_n$ is a sequence in $\mathbb{R^n}$ that converges to $x*$ and $p=1\Rightarrow 0<c<1$. I have problems to prove the equality and use the $\liminf$ operator i.e. to justify. I also dont know ...
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1answer
113 views

Showing a biconditional statement about function lim sups in $\Bbb R^n$, and codifying the intuition into a proof

$ \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\BB}{\mathcal{B}} \newcommand{\ve}{\varepsilon} \newcommand{\para}[1]{\left( #1 \right)} \newcommand{\set}[1]{\left\{ #1 \right\} }...
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2answers
55 views

For $0 \leq a_n, b_n$: Is $\varliminf a_n \varlimsup b_n \leq \varlimsup a_n b_n $ true?

The sequences are assumed to be bounded! I suspect it is, since $\varliminf a_n + \varlimsup b_n \leq \varlimsup (a_n+b_n)$ is. But I have not managed to prove it. I thought it possible to mirror the ...
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1answer
28 views

Convergence of signed measures implies inequality between variations

I'm trying to tackle Junghenn's Principles of Real Analysis' exercise 5.11: Let $\mu, \mu_1, \mu_2, \dots$ be signed measures on $(X, \mathfrak{F})$ with $\mu_n \to \mu$. Show that $\mu^{\pm}(E) \leq ...
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1answer
32 views

Does this liminf characterization hold true?

Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function such that $$0<a:=\liminf_{|t|\to\infty} tf(t) <\infty.$$ I am interested in writing the above relation by using the definition. On by ...
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0answers
27 views

Limit supremum of x/n

To be more specific, the question is asking for $$\lim_{n\rightarrow\infty}\sup\{\frac{x}{n}|x\in\mathbb{R}\}$$ I'm not sure if the dominating variable is $x$ or $n$.
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2answers
51 views

Difference between lim and lim sup of a function with a so called “blow up”?

Let $f:[0,\infty)\rightarrow\mathbb{R}$ be a function with a "blow up" in finite time i.e. $$\limsup\limits_{t\uparrow T_{max}}|f(t)|=\infty.$$ I don't unterstand the difference between lim ...
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1answer
46 views

How to compute and think intuitively about $\limsup$ and $\liminf$ of the below function?

$$x_n = \frac{n^2}{1+n^2} \cos \frac{n \pi}{16} $$ Could someone please let me know how to derive the limits in this case? I can think — with using l'Hopital rule the limit of $\frac{n^2}{1+n^2} $ is $...
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0answers
84 views

Determine the limit $\lim_{ ( x , y ) \to ( 0 , 0 )} (1 − x ^2 − y^ 2)/( x^ 2 + y ^2)$

Question : Determine whether the limit exists for the following. If so, find their value. $\lim_{ ( x , y ) \to ( 0 , 0 )} (1 − x ^2 − y^ 2)/( x^ 2 + y ^2)$ Normally we solve this by either cancelling ...
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22 views

Interpret $\limsup_{n \to\infty} K_n := \left\{ x \in X \;|\; \liminf_{n\to\infty} d(x,K_n) = 0 \right\}$

Let $(K_n)_{n \in \mathbb N}$ be a sequence of subsets of a metric space $(X,d)$. We say the subset $$ \limsup_{n \to\infty} K_n := \left\{ x \in X \;|\; \liminf_{n\to\infty} d(x,K_n) = 0 \right\}, $$ ...
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1answer
18 views

Limit Supremum of Sequence Created by Interleaving 2 Bounded Sequences

I'm studying for an upcoming departmental exam, and this was one of the questions given on a previous year's exam. Let $(a_{n})$ and $(b_{n})$ be real, bounded sequences, and let $(c_{n})$ be the ...
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0answers
36 views

Limit of $ \frac{1}{2}\frac{\sin^2(\omega t/2)}{(\omega/2)^2}$

I'm trying to prove that $\lim_{t\rightarrow \infty} \frac{1}{2}\frac{\sin^2(\omega t/2)}{(\omega/2)^2} = \pi \delta(\omega) t $ Any thoughts on how could I do that? I thought about mapping it to the ...
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1answer
29 views

Liminf of Pointwise Norms of a Weakly Convergent Sequence

Let $X_1, X_2, \cdots$ be a sequence of $p$-integrable $\mathbb{R^d}$ valued random variables. Assume that $X_n$ converges $0$ weakly, then can we say that $r(\omega) = liminf\{ |X_1 (\omega)|, |X_2 (\...
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1answer
129 views

On the proof of Thm.3.50 in Baby Rudin

I have a question on the proof of Thm.3.50 in Rudin's Principles of Mathematical Analysis. This theorem is about the convergence of the Cauchy product of two infinite series. For convenience, I write ...
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0answers
28 views

Proving limsup of Brownian motion is not zero a.s. using $\mathbb{P}(B_1\leq0,\sup_{t\geq0}B_{t+1}−B_1 =0)$

Consider $M_t:=\sup\limits_{t\geq0}B_t$. I have managed to prove that the law of $M_t$ is concentrated on $\{0,\infty\}$ using the distribution of scaled Brownian motion. However, I am asked to prove $...
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2answers
46 views

If $\limsup_{n\to\infty}a_n<\infty$, then $\limsup_{n\to\infty}(-a_n)>-\infty$?

Assume $(a_n)$ is a positive sequence. I stuck to find if the following always true? If $\limsup_{n\to\infty}a_n<\infty$, then $\limsup_{n\to\infty}(-a_n)>-\infty$.
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1answer
84 views

Inequality relating the Limit Superior and Limit Inferior of a bounded sequence

I'm doing exercise 2.4.7 in the book Understanding Analysis by Stephen Abbott. I'd like to ask, how to go about part (d) of the proof (please don't give away the ...
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1answer
20 views

The divergence of a series implies the divergence of limit superior of absolute value of sums of random variables

I'm here for another problem. I need to show the following: If $\{X_n\}$ is a sequence of independent random variables, and $S_n=\sum_{k=1}^{n}X_k$ then $$\limsup_{n}|S_n|=\infty \quad a.s. \text{ if }...
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1answer
40 views

Does the following identity involving limits and limsup/liminf hold?

Suppose I have three sequences $(a_n)_{n\ge1}$, $(b_n)_{n\ge1}$ and $(c_n)_{n\ge1}$ on the real line which satisfy: $a_n = b_n + c_n$ The sequence $a_1,a_2,\ldots$ converges (in R), and all elements ...
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2answers
105 views

Double limit question

Let $f: X \times Y \rightarrow \mathbb{R}$ be a continuous function where $X \subseteq \mathbb{R}$ and $Y \subseteq \mathbb{R}$. Suppose that $(x_{n},y_{n}) \rightarrow (x,y)$ in $\mathbb{R}^{2}$, $(...
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1answer
44 views

Do $\limsup$ and $\liminf$ are the actual bounds of a sequence? [closed]

Can you say that $\limsup a_n$ is the upper bound of the sequence $a_n$ and $\liminf a_n$ is the lower bound of $a_n$?
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0answers
33 views

Proving equivalence of definitions of limit superior

I tasked myself with proving the equivalence of these two definitions of limit superior for a bounded sequence: $ (1) \ \overline{\lim} a_n = \sup \{ x \in \mathbb{R} : x < a_n \ \text{ for ...
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1answer
18 views

Showing non-differentiability of a function at $0$

Starting from a function $f:t\mapsto f(t)$ such that: $f(0)=0\tag{1}$ $\liminf\limits_{t\to\infty} \sqrt{t}f\left(\dfrac{1}{t}\right)=-\infty\tag{2}$ $\limsup\limits_{t\to\infty} \sqrt{t}f\left(\...
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1answer
45 views

Prove or give a counterexample of limsup and liminf [closed]

Lets $a_n$ and $b_n$ be two sequence that $0\leqslant a_n \leqslant 1$ and $0\leqslant b_n\leqslant 1$ for each $n$. Prove or give a counterexample that: a) if $\displaystyle\liminf_{n\to\infty} ...
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3answers
415 views

Why does the limit definition of e fail?

I'm trying to calculate the following limit: $$\lim_{x\to +\infty} \frac{(1+\frac{1}{x})^{x^2}}{e^x}$$ I tried to use the fact that $e^x = \lim_{x\to +\infty}(1+\frac{1}{x})^{x}$, but this gives $\...
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0answers
19 views

If LimInf An > 0, then from a certain point on the sequence, all the terms are positive

Given a general sequence An, and also given that LimInf An > 0, prove that from a certain point on the sequence is strictly positive. Im having troubles approaching this question, I saw by the ...
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0answers
18 views

Root test Supersedes Ratio test

This is problem 65 a) on page 208 of Pugh's Real Mathematical Analysis. Show that \begin{equation} \limsup\limits_{n\rightarrow \infty}|a_n|^{\frac{1}{n}}=\rho \\ \implies \\ \limsup\limits_{n\...
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0answers
18 views

$\limsup$ for the sum of random variables

Let $(a_n)_{n \in \mathbb{N}}$ be a sequence of non-negative real numbers such that $\lim_{n \to \infty} a_n=\infty$ and let $(X_n)_{n \in \mathbb{N}}$ be a sequence of real-valued random variables. ...
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1answer
40 views

The limit of a limsup of a nonnegative sequence is 0

I am currently looking at a result that states, for some nonnegative function $g: \mathbb{R} \times \mathbb{R} \to [0,1]$, there exists a constant $c> 0$ such that \begin{align} \lim_{k \to \infty} ...
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0answers
92 views

If $u^*(x) = \lim_{r \to 0} \sup_{B_r(x)} u$ then $u^*$ is the smallest upper semicontinuous funtion which is greater then $u$

In the context of vicosity solutions for fully nonlinear PDE of second order (I don't know first order theory) it is useful to define the upper semicontinuous envelope of $u : \Omega \longrightarrow \...
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2answers
66 views

Dini derivatives of Dirichlet function

I'm hoping someone can verify my answers. I have that for $x\in\mathbb{Q}$ we have $$D^+f(x)=\limsup_{h\to 0^+}\frac{f(x+h)-f(x)}{h}=\limsup_{h\to 0^+}\frac{f(x+h)-1}{h}$$ and since $\mathbb{Q}$ is ...

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