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.
1,612
questions
0
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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 ...
1
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0answers
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 $$\limsup_{d\in D} a_d = +\infty \quad \text{and} \quad \lim_{d\in D} b_d = +\infty.$$
(...
0
votes
1answer
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
$$a=\sup\{ a_n | n \in \mathbb{N} \...
0
votes
2answers
44 views
Construct a sequence of functions in $L^p(\mathbb{R})$ such that $f_n$ converges to zero in $L^p$. [closed]
I'm trying to find a sequence of functions $f_n\in L^p(\mathbb{R})$, with $1\leq p < \infty$ such that $\lVert f_n \rVert_{L^p}\to 0$. Moreover for all $x\in \mathbb{R}$ the sequence satisfy two ...
0
votes
1answer
23 views
Existence of subsequences $a_{n_k}$ that converges to a value between $\liminf a_n$ and $\limsup a_n$
I am stuck on this question:
Suppose $a_n$ is a sequence with $\liminf a_n=s$ and $\limsup a_n=t$ where $s$ and $t$ are real number. Then there must exist some subsequence $a_{n_k}$ that converges to $...
0
votes
1answer
34 views
If lim inf $a_n$ = $\infty$ and lim sup $a_n$ = $-\infty$, $a_n$ diverges?
I know that a sequence is a convergent sequence if and only if it is a Cauchy sequence so lim inf $a_n$ = $\lim sup a_n$.
Then, if $\liminf a_n = \infty$, then can we conclude $\lim a_n=\infty$ ?
What ...
2
votes
1answer
33 views
How to “explain in words” this expression involving liminf
Let $\Omega\subset\mathbb{R}^N$ and $F:\Omega\times\mathbb{R}\to\mathbb{R}$ be a function and consider the expression
$$ \liminf_{| u|\to+\infty} \frac{F(x, u(x))}{|u|^p +| u|^q} <0,$$
for some $p, ...
1
vote
1answer
40 views
Why is lim sup necessary
I have hard time understanding why the concept of lim superior is necessary, for example the ratio test. There are two ways, it seems of phrasing this:
(Abbott, Understanding Analysis p.78) Given $\...
0
votes
1answer
31 views
Comparing two events
Let $X_n$ be a sequence of real random variables, $X$ a real random variable and $\varepsilon > 0$
Let $A_n$ be the event $ \{ | X_n - X | \geq \varepsilon \} $ and $B_n$ the event $ \{ \sup_{k \...
0
votes
0answers
36 views
How to prove $\limsup_{n\to \infty} \sqrt[n]{|a_n|}=\limsup_{n\to \infty} \sqrt[n]{|a_{n+1}-a_n|}$ if $\{a_n\}$ diverges [duplicate]
$\{a_n\}$ is a diverging sequence.
I have proven that $\limsup_{n\to \infty} \sqrt[n]{|a_{n+1}-a_n|}\geq1$ by contrapositive.
Then, we need to prove $\limsup_{n\to \infty} \sqrt[n]{|a_n|}=\limsup_{n\...
2
votes
0answers
29 views
Alternative proof of generalized Borel-Cantelli without the use of the finite increment martingale limit property
I'm working through a proof outline of the generalized Borel-Cantelli lemmas. Following the outline, I've shown that for $A_k\in\mathcal F_k$ and
$$X_n=\sum_{k=1}^n 1_{A_k}, Y_n=\sum_{k=1}^n P(A_k\...
2
votes
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:
$$x_n = \cos\left(\frac{n\pi}{4}\right) + \sin\left(\frac{n\pi}{2}\right) + 2^{-n} + (-1)^n$$...
1
vote
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 ...
1
vote
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 ...
2
votes
0answers
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))...
2
votes
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{...
1
vote
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 $\mathbb{R}^d$).
...
1
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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 \...
1
vote
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
$$A:=\{r\geq 0: \exists M\in\mathbb{R}[\forall n\in \mathbb{N} [\, |r^n a_n|<M\ ]] \}$$
$$B := \{ r\geq 0 :\sum_n |r^n ...
0
votes
0answers
16 views
$\Vert x_{n +1}-x^*\Vert\leq c\Vert x_n-x^*\Vert ^p\Rightarrow p=\liminf\frac{\ln\Vert x_{n+1}-x^*\Vert}{\ln\Vert x_n-x^*\Vert}$
$x_n$ is a sequence in $\mathbb{R^n}$ that converges to $x*$ and $p=1\Rightarrow 0<c<1$. I have problems to prove the equality and use the $\liminf$ operator i.e. to justify. I also dont know ...
6
votes
1answer
113 views
Showing a biconditional statement about function lim sups in $\Bbb R^n$, and codifying the intuition into a proof
$
\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\BB}{\mathcal{B}}
\newcommand{\ve}{\varepsilon}
\newcommand{\para}[1]{\left( #1 \right)}
\newcommand{\set}[1]{\left\{ #1 \right\} }...
1
vote
2answers
55 views
For $0 \leq a_n, b_n$: Is $\varliminf a_n \varlimsup b_n \leq \varlimsup a_n b_n $ true?
The sequences are assumed to be bounded!
I suspect it is, since $\varliminf a_n + \varlimsup b_n \leq \varlimsup (a_n+b_n)$ is. But I have not managed to prove it.
I thought it possible to mirror the ...
0
votes
1answer
28 views
Convergence of signed measures implies inequality between variations
I'm trying to tackle Junghenn's Principles of Real Analysis' exercise 5.11:
Let $\mu, \mu_1, \mu_2, \dots$ be signed measures on $(X, \mathfrak{F})$ with $\mu_n \to \mu$. Show that $\mu^{\pm}(E) \leq ...
1
vote
1answer
32 views
Does this liminf characterization hold true?
Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function such that
$$0<a:=\liminf_{|t|\to\infty} tf(t) <\infty.$$
I am interested in writing the above relation by using the definition. On by ...
0
votes
0answers
27 views
Limit supremum of x/n
To be more specific, the question is asking for $$\lim_{n\rightarrow\infty}\sup\{\frac{x}{n}|x\in\mathbb{R}\}$$
I'm not sure if the dominating variable is $x$ or $n$.
0
votes
2answers
51 views
Difference between lim and lim sup of a function with a so called “blow up”?
Let $f:[0,\infty)\rightarrow\mathbb{R}$ be a function with a "blow up" in finite time i.e. $$\limsup\limits_{t\uparrow T_{max}}|f(t)|=\infty.$$
I don't unterstand the difference between lim ...
1
vote
1answer
46 views
How to compute and think intuitively about $\limsup$ and $\liminf$ of the below function?
$$x_n = \frac{n^2}{1+n^2} \cos \frac{n \pi}{16} $$
Could someone please let me know how to derive the limits in this case?
I can think — with using l'Hopital rule the limit of $\frac{n^2}{1+n^2} $ is $...
1
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0answers
84 views
Determine the limit $\lim_{ ( x , y ) \to ( 0 , 0 )} (1 − x ^2 − y^ 2)/( x^ 2 + y ^2)$
Question : Determine whether the limit exists for the following. If so, find their value.
$\lim_{ ( x , y ) \to ( 0 , 0 )} (1 − x ^2 − y^ 2)/( x^ 2 + y ^2)$
Normally we solve this by either cancelling ...
0
votes
0answers
22 views
Interpret $\limsup_{n \to\infty} K_n := \left\{ x \in X \;|\; \liminf_{n\to\infty} d(x,K_n) = 0 \right\}$
Let $(K_n)_{n \in \mathbb N}$ be a sequence of subsets of a metric space $(X,d)$. We say the subset
$$
\limsup_{n \to\infty} K_n := \left\{ x \in X \;|\; \liminf_{n\to\infty} d(x,K_n) = 0 \right\},
$$
...
0
votes
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 ...
0
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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 ...
0
votes
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 (\...
0
votes
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 ...
2
votes
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 $...
1
vote
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$.
1
vote
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 ...
0
votes
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 }...
0
votes
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 ...
0
votes
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 $\mathbb{R}^{2}$,
$(...
0
votes
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 $a_n$?
0
votes
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 ...
0
votes
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(\...
0
votes
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} ...
5
votes
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 $\...
0
votes
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 ...
0
votes
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\...
0
votes
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. ...
0
votes
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} ...
1
vote
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 \...
1
vote
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 ...
