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 $\lim \inf (a_n)+\lim \inf (b_n) \leq \lim \inf (a_n+b_n)$ using subsequence

Let $\left(a_{n}\right)_{n=1}^{\infty},\left(b_{n}\right)_{n=1}^{\infty}$ be bounded sequences. Prove that $\underset{n\rightarrow\infty}{\lim\inf}\left(a_{n}\right)+\underset{n\rightarrow\infty}{\lim\...
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Coinciding Limsup and limit

Let $(x_n)_{n \geq 1}$ be a sequence of real numbers. Suppose that we are able to show that for a fixed number $m$, $(y_n)_{n \geq 1}:= (x_{n+m})$ and we know that $\lim_{n\to\infty}(y_n)=x$ for some ...
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An inequality for fixed points of dynamical systems

Suppose $f:X\to X$ is a continuous map such that each iterate has finitely many fixed points. Define $h(f) = \limsup_{k\to \infty} \frac{\log \#Fix(f^k)}{k}$ where $\#Fix(f^k)$ is the number of fixed ...
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An exercise on counting fixed points from Milnor

I am working on the following exercise (Problem 6-b, pg. 6-22) from Milnor's Dynamical Systems notes: Problem 6-b Let $f$ be any self map such that each iterate has only finitely many fixed points. ...
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Find $a, b \in \mathbb{R}$ such that power series $ \sum_{n=1}^{\infty} \frac{\arctan n^a}{n^b} x^n$ converges.

Find $a, b \in \mathbb{R}$ such that power series $$ \sum_{n=1}^{\infty} \frac{\arctan n^a}{n^b} x^n$$ converges. I had problem in finding radius of convergence, more precisely I don't know how to ...
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Prove the inequality in Banach space

Let $X$ be a Banach space and let $(x_n)_{n=1}^\infty$ be a sequence in $X$ such that, for some $x \in X$, $$ \lim_{n \rightarrow \infty} \ell(x_n) = \ell(x) \text{ for all bounded linear functional }...
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Symmetric Difference Question

Let {Dn} be a sequence of sets defined by D1=E1, D2=D1 Δ E1,.....,Dn=Dn-1 Δ En; n=2,3,..... Show that {Dn} converges iff lim En= φ.
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How do we prove the given definitions of $\liminf A_{n}$ and $\limsup A_{n}$ are equivalent?

I have to prove two similar identities involving the limit superior/inferior of a sequence of events in the event space but I'm not sure how to proceed. Here are the identities: \begin{align*} \...
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limsup of fraction ratio to limsup

Could one generalize this result in the following way? Let $(a_n)_{n \in \mathbb{N}}$ be a nondecreasing sequence of positive real numbers. Then for any strictly increasing subsequence $n_k$ with $\...
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Right and left side properties of limsup

let $f:\mathbb{R}\to\mathbb{R}$ be a real function such that limsups from the left and right are finite. Let $g(x):=\limsup_{t\to x+}f(t)$. Is it true that $\limsup_{s\to x-}f(s)=\limsup_{s\to x-}g(s)$...
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Convergence of $\sum_{n=1}^{\infty}\frac{1}{n^{\gamma_n}}$

Suppose $\lbrace\gamma_n\rbrace_{n\in\mathbb{N}}$ is a sequence of real numbers such that $\gamma_n>1$ for all $n$ and $\liminf\gamma_n=1$. Under which conditions (if any) does $$ \sum_{n=1}^{\...
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Find $\limsup (a_n)^{\frac{1}{n}}$ given that $\limsup a_n =R>0$, and $a_n\geq 0$ for each $n\in\mathbb{N}$

I'm trying to find if there's a relation between $\limsup (a_n)^{\frac{1}{n}}$ and $\limsup a_n $, given that $\limsup a_n =R>0$ and $a_n\geq 0$ for each $n\in\mathbb{N}$. I've found some examples ...
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Limit superior and limit inferior of sets

I have searched for the answer in wikipedia and math stackexchange. However, I do not have any background in real analysis and all the answers seem very complicated to me to understand. I am wondering ...
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How to estimate numerically a lim sup/lim inf

Suppose I have a sequence of random variables $X_n$ s.t. $\lim_{n\to\infty}X_n=c$ a.s. For example, we could take the $X_n$ to be the sample mean from some well-behaved distribution. If I have a way ...
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How do I rewrite this probability statement using $\limsup$?

I have the following problem. We have $Y_n$ independent random variables defined on $(\Omega, F, \Bbb{P})$ s.t. $$\Bbb{P}(Y_n=1)=p~~~\Bbb{P}(Y_n=0)=1-p$$ for $p\in [0,1]$. We define $A_0=0$ and $A_n=\...
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5 votes
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Proving $ 2xf(x)+g(x)^2 \leq M(1+\vert x \vert^2) $

Let $f,g \in C^1$ and assume there exists $k,K>0$ s.t. for all $x \in \mathbb{R}$ $$ k\leq\vert g(x)\vert \leq K(1+\vert x \vert). $$ Assume furthermore that $$ \limsup_{\vert x \vert \to \infty} \...
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2 answers
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$\limsup_{x \to \infty} f(x) < 0 \implies \exists c,K>0$ s.t. $x > K \Rightarrow f(x) < -c$?

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ and suppose $\limsup_{x \to \infty} f(x) < 0$. Is is it then true that there exists $c,K>0$ s.t. for $x > K$ we have that $f(x) < -c$? My ...
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  • 195
1 vote
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Changing limit and sup

Let be $ A $ a set of convergent sequences. Suppose $$ b_n:=\sup\{a_n:(a_n)_{n\in \mathbb{N}}\in A\} $$ does exist for all $ n\in \mathbb{N} $. Then for $ (b_n)_{n\in \mathbb{N}} $ follws $$ \lim\...
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How $\limsup_{n\to\infty}\left[\frac{a_n}{n^\varepsilon}\right]^{1/\log\log n}=e^{1+\varepsilon}\implies a_n=O(n^\varepsilon\log^{1+\varepsilon}n)$?

Question Let $(a_n) \subseteq \mathbb R_+$ be a sequence of non-negative numbers such that $$ \limsup_{n \to \infty} \left[ \frac{a_n}{n^\varepsilon} \right]^{1/\log \log n} = e^{1+\varepsilon} $$ for ...
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Can we use power mean to generalize min and max for complex numbers?

Power mean $M_p(a,b)$ of order $p \in \mathbb{R}$ for a pair $(a,b) \in \mathbb{R}^+$ is defined as $M_p(a,b)= \Big(\frac{a^p+b^p}{2}\Big)^{\frac{1}{p}}$. For example $p = 1$ gives arithmetic mean, ...
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How can I solve this without L'Hôpital's rule or Taylor series? [closed]

How can I solve this limit without L'Hôpital's rule or Taylor series?$$\lim_{x\to -1}\frac{\sin(x^3-x)}{x+1}.$$ I was trying to solve this limit but I'm stuck when I multiply it by conjugate of the ...
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1 vote
0 answers
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Positive Measure of Limsup of Subsequence

Given a measure space $\Omega$, a sequence of measurable functions $(f_n : \Omega \rightarrow \mathbb{R})_{n=1}^\infty$ and $r \in \mathbb{R}$, such that: $\{ \omega \in \Omega : \limsup_{n \...
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5 votes
1 answer
121 views

Minimal assumption on points of discontinuity of $f$ so that $\liminf_{\epsilon \to \infty} \frac{f(x)}{f(x+\frac{t}{\epsilon} )}=1$

Suppose we are given a function $f(x)$. We want to show the following claim: \begin{align} \liminf_{\epsilon \to \infty} \frac{f(x)}{f(x+\frac{t}{\epsilon} )}=1, \end{align} almost everywhere $(x,t)$ ...
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set limits, lim sup and lim inf

Let $X$ be continuum and $\{C_i\}$ a sequence of compacts set in $X$ then $\limsup C_i$ and $\liminf C_i$ is compacts. where $(C_i)_{i=1}^{\infty} \subseteq \mathcal{P}(X)$ and $\liminf C_i =\{x \in ...
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1 answer
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"Properties" of the definitions of limit superior and inferior

I was working on a proof about measure theory, where I was asked to show that for any sequence of subsets $\left(A_{n}:n\in \mathbb{N}\right)$ of some set $X$, if we created another sequence of ...
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2 votes
2 answers
59 views

Find $\liminf _{n \to \infty}A_n $ and $\limsup_{n \to \infty}A_n $

Define $$A_n= \begin{cases} \left ( \frac{1}{2}-\frac{1}{2n},1+\frac{1}{2n}\right )& \text{ if n is odd } \\ \left ( \frac{1}{2n},\frac{3}{4}-\frac{1}{2n} \right )& \text{ if n is even } \...
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1 vote
0 answers
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Whether the formula $\limsup\frac{\max\{x_1, x_2,...,x_n\}}{a_n}=\limsup\frac{x_n}{a_n}$ holds?

Here, $a_n \uparrow \infty$, $x_n$ is a sequence, it can take negative or positve value. Does the following formula hold? If so, can you provide a proof? \begin{equation} \limsup\frac{\max\{x_1, x_2,.....
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  • 409
3 votes
1 answer
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Limit of flipping coins with $\limsup_{n\rightarrow\infty}\frac{N_n}{\log n}=\frac{1}{\log 2}$

We flip a fair coin repeatedly and independently. Let $N_n$ be the number of consecutive heads beginning from the $n^{th}$ flip. (For example, $N_n =0$ if the $n^{th}$ flip is a tail, and $N_n =2$ if ...
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  • 804
0 votes
2 answers
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What is the superior limit of this sequence?

So I've been given this sequence: $a_n:=\cos\left(\frac{2n^2+1}{3n}\pi\right)$ and we are asked to determine its superior limit. My first thought was that I can probably write it (due to continuity of ...
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1 answer
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Statements True or False? Real Analysis

(a) If the set $S$ of subsequential limits of $(x_n)$ is empty, then $(x_n)$ is unbounded. (b) If $a$ is an eventual upper bound for $(x_n)$ and $b$ is an eventual upper bound for $(b_n)$, then $a-b$ ...
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  • 409
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1 answer
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Finding a sequence satisfing this $\liminf$ equation

Let $(X,d)$ be a metric space and let $f_j : X \to [-\infty , +\infty ]$ for all $j\in \mathbb{N}$. Observe that since $X$ is metric, compactness is equivalent to sequential compactness. Problem: If ...
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0 votes
0 answers
60 views

Finding a "recovery" sequence for $f$

Let $(X,d)$ be a metric space and let $f:X\to [-\infty, +\infty] $. Now we fix a $x\in X$ and we want to find a sequence $(x_j)\subset X$ convergint to $x$ such that $$\limsup_{j\to +\infty} f(x_j) \...
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  • 1,148
1 vote
1 answer
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$\lim\sup$ and $\lim\inf$ of a sequence of independent random variables with two states

Let $(X_n)_{n\geq1}$ be a sequence of random variables such that $$ \mathbb{P}(X_n= n^{2/3})=1-\mathbb{P}(X_n=0)=\frac{1}{3n}$$ Find $\lim\sup X_n$ and $\lim\inf X_n$. We know that \begin{align} \{\...
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0 answers
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let $(a_n)$ be the sequence $(1, 0, 2, 0, 3, 0, . . .)$. Find $\lim \inf$, $\lim \sup$ of $(a_n)$ and $(b_n)$ where $b_n= (a_1+a_2+...+a_n)/n$

Finding $\lim \inf$ and $\lim \sup$ of $a_n$ is easy. Without using any theorems one can also see $\lim \sup$ of $b_n$ is $+\infty$ as every even term is of form $\frac{n+2}{8}$ but how does one find $...
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1 vote
2 answers
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Is zero the liminf of the norm of a weakly convergence sequence to zero in HIlbert spaces?

Let $X$ Hilbert and ${x_n} \subset X$ such that $x_n \rightharpoonup x \in X$ (where this is the weak convergence in Hilbert spaces). Now of course the following doesn't happen in general $$\lim_n |...
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2 answers
49 views

"Change of variables" in $\limsup_{x\rightarrow\infty}f(x+t)$

Let $f:(0,\infty)\rightarrow \mathbb{R}$ be a real-valued function, such that for some $t,C>0$, \begin{equation} \limsup_{x\rightarrow\infty} f(x+t)\leq C \end{equation} Is it also true that $\...
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Lower Semi Continuous Function Multiplied by a Bounded Positive Sequence.

Let $A:\mathcal{P}_2(\mathbb{R})\to \mathbb{R}$, be a lower semi-continuous functional on the space of Borel probability measures over the real line. Lower semi-continuous here means with respect to ...
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2 votes
0 answers
37 views

From sequence $m_n = o(f(n))$ to function $m(x) = o(f(x))$?

Question Let $f : \mathbb R_+ \to \mathbb R_+$ be a continuous, non-decreasing function with $f(x) \to \infty$ as $x \to \infty$. Suppose we have a non-decreasing sequence $(m_n) \subset \mathbb N$ ...
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1 vote
1 answer
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Proving that $\limsup_{h \to 0^{+}} \frac{F(x + h) - F(x)}{h}$ is measurable if $F$ is continuous

I am trying to write up a detailed proof for the following statement: Let $F: \mathbb{R} \rightarrow \mathbb{R}$ be continuous. Then $$D^{+}(F)(x) := \limsup_{h \to 0^{+}} \frac{F(x+h) - F(x)}{h}$$ ...
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  • 816
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Limsup & liminf of $\exp(-\cos(n))$

Let $a_n = \exp(-\cos(n))$. I need to find $\limsup_{n \to \infty}(a_n)$ and $\liminf_{n \to \infty}(a_n)$. By applying the definition of $\limsup$ I got that: $$\limsup_{n \to \infty}(a_n) = \lim_{n \...
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1 answer
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Set of points where lower left Dini derivative dominates the upper right Dini derivative is countable

I am self studying analysis and came across a theorem that states that for any real valued function $f$, the set of points where the upper right derivative of $f$ is strictly less than the lower left ...
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  • 97
2 votes
1 answer
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Is $\liminf f(x)$ always unique?

Searching for the question doesn't seem to give any relevant answers on stack exchange or on google. I know that $\liminf_{n \rightarrow \infty} f(x_n)$ is always unique, but I haven't seen a proof ...
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  • 1,430
4 votes
1 answer
182 views

Understanding what $\displaystyle \lim\sup_{x \to a, x \in E} f(x)$ means

I recently learned of the following extension to the definition of a limit: Let $S \subset \mathbb{R}$, let $f: S \rightarrow \mathbb{R}$, let $a$ be a limit point of $S$, and let $L \in \mathbb{R}$. ...
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0 votes
1 answer
61 views

Could lim sup not be in tail event?

I know that $\lim \sup A_n$ is always a tail event. E.g. from here. But this was asked in an exam a few years ago - Prove of disprove: Let $X_1, X_2, ... $ be a series of positive R.V.; Let $S_n = ...
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(When) do students typically learn about limit superior and inferior?

I've been taking the Khan AP Calculus BC course, and I also have been following through with pretty extensive calculus notes found here. In neither of these (or when I learned Calc in high school) ...
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1 answer
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Does the limit : $\lim _{x\to \infty }\frac{\ln x^{\frac{1}{3}}}{\sin x}$ exist

I tried to evaluate the limit $$\lim _{x\to \infty }\frac{\ln x^{\frac{1}{3}}}{\sin x}$$ but came to the conclusion that the limit would not exist as $\sin x$ changes sign too frequently so obviously ...
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  • 7,082
0 votes
1 answer
63 views

A nonnegative, integrable, Lipschitz function $f$ satisfies $\lim \inf_{n \rightarrow \infty} \sqrt{n}f(n) = 0$

Let $f$ be a nonnegative integrable and Lipschitz function in $\mathbb{R}$ with Lipschitz constant $C$. Prove that $\lim \inf_{n \rightarrow \infty} \sqrt{n}f(n) = 0.$ The issue I have with this ...
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Show that $\limsup_{x\to p}f(x)$ is the unique element, which satisfies a certain condition.(follow-up question )

This is follow up question of the following two questions I asked Sequencial chracterisation of $\limsup f(x)$ Show equivalence of definitons of $\limsup f(x)$ I would like to show that there is $\...
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2 votes
0 answers
69 views

Show equivalence of definitons of $\limsup f(x)$

This is a follow up question to the following question I asked. Sequencial chracterisation of $\limsup f(x)$ . I would like to pove that is equivalent to Proof attempt "$\Leftarrow$" Let $\...
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Supremum, limit and limsup

Let {$a_n$} and {$b_n$} be two sequences of real numbers such that {$b_n$} converges to L. It is known that the supremum S of the set {$a_n$: n $\in$ N} exists and is not equal to any of the $a_j$'s. ...
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