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.

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

The limit superior and the limit inferior of a sequence $a_1,a_2,a_3,\ldots$ of real numbers are real numbers, $+\infty$ or $-\infty$. More precisely, the limit superior is the limit of the sequence $\sup_{n\in\mathbb N}${$a_n,a_{n+1},a_{n+2},\ldots$} and the limit inferior is the limit of the sequence $\inf_{n\in\mathbb N}${$a_n,a_{n+1},a_{n+2},\ldots$}. Each sequence of real numbers has one and only one limit superior and one and only one limit inferior. They are equal if and only if the sequence has a limit.

The limit superior and the limit inferior of a sequence $S_1,S_2,S_3,\ldots$ of sets are respectively the sets $\limsup_nS_n=\bigcap_{i=1}^{+\infty}\bigcup_{j=i}^{+\infty}S_j$ and $\liminf_nS_n=\bigcup_{i=1}^{+\infty}\bigcap_{j=i}^{+\infty}S_j$.

Use this tag along with or as is found appropriate. For questions concerning the limsup/inf of sets, please add the tag. For questions involving abstract partial orders, use also the tag.

For questions concerning the evaluation and other properties of limits, use the tag.

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lim sup and lim inf of sequence of sets.

I was wondering if someone would be so kind to provide a very simple explanation of lim sup and lim inf of s sequence of sets. For a sequence of subsets $A_n$ of a set $X$, the $\limsup A_n= \bigcap_{N=1}^\infty \left( \bigcup_{n\ge N} A_n \right)$…
Comic Book Guy
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Can someone clearly explain about the lim sup and lim inf?

Can some explain the lim sup and lim inf? In my text book the definition of these two is this. Let $(s_n)$ be a sequence in $\mathbb{R}$. We define $$\lim \sup\ s_n = \lim_{N \rightarrow \infty} \sup\{s_n:n>N\}$$ and $$\lim\inf\ s_n =…
eChung00
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Prove $\limsup\limits_{n \to \infty} (a_n+b_n) \le \limsup\limits_{n \to \infty} a_n + \limsup\limits_{n \to \infty} b_n$

I am stuck with the following problem. Prove that $$\limsup_{n \to \infty} (a_n+b_n) \le \limsup_{n \to \infty} a_n + \limsup_{n \to \infty} b_n$$ I was thinking of using the triangle inequality saying $$|a_n + b_n| \le |a_n| + |b_n|$$ but the…
hyg17
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What is limit superior and limit inferior?

I've looked at the Wikipedia article, but it seems like gibberish. The only thing I was able to pick out of it was the concept of infimum (greatest lower bound) and supremum (least upper bound), as I had learned them previously in an intro discrete…
Mirrana
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Interpretation of limsup-liminf of sets

What is an intuitive interpretation of the 'events' $$\limsup A_n:=\bigcap_{n=0}^{\infty}\bigcup_{k=n}^{\infty}A_k$$ and $$\liminf A_n:=\bigcup_{n=0}^{\infty}\bigcap_{k=n}^{\infty}A_k$$ when $A_n$ are subsets of a measured space $(\Omega,…
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Two definitions of $\limsup$

Here are two equivalent definitions of $\limsup_{n\rightarrow\infty} a_n$: Let $u_n=\sup\{a_n, a_{n+1}, a_{n+2},\ldots\}$. Then $$\limsup_{n\rightarrow\infty} a_n = \lim_{n\rightarrow\infty} u_n = …
angryavian
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How would you evaluate $\liminf\limits_{n\to\infty} \ n \,|\mathopen{}\sin n|$

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 of $\pi$. Since $\mu$ is not known, I will split the…
Τίμων
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Properties of $\liminf$ and $\limsup$ of sum of sequences: $\limsup s_n + \liminf t_n \leq \limsup (s_n + t_n) \leq \limsup s_n + \limsup t_n$

Let $\{s_n\}$ and $\{t_n\}$ be sequences. I've noticed this inequality in a few analysis textbooks that I have come across, so I've started to think this can't be a typo: $\limsup\limits_{n \rightarrow \infty} s_n + \liminf\limits_{n \rightarrow…
emka
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Sequence converges iff $\limsup = \liminf$

I want to prove that a sequence of real numbers $\{s_n\}$ converges to $s$ if and only if $\limsup_{n \to \infty} s_n = \liminf_{n \to \infty} s_n = s$. Here are my definitions: For any sequence of real numbers $\{s_n\}$, let $E$ be the set of all…
jamaicanworm
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limit inferior and superior for sets vs real numbers

I am looking for an intuitive explanation of $\liminf$ and $\limsup$ for sequence of sets and how it corresponds to $\liminf$ and $\limsup$ for sets of real numbers. I researched online but cannot find a good comparison. Any link, reference or…
user957
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A subadditive bijection on the positive reals

Question. Does there exist a subadditive bijection $f$ of the positive reals $(0,\infty)$ such that $$ \liminf_{x\to 0^+}f(x)=0 \,\,\,\text{ and }\,\,\,\limsup_{x\to 0^+}f(x)=1\,? $$ Ps1. I guess the answer is no. However, it is affirmative if we…
Paolo Leonetti
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Can someone explain the Borel-Cantelli Lemma?

I’m looking for an informal and intuitive explanation of the Borel-Cantelli Lemma. The symbolic version can be found here. What is confusing me is what ‘probability of the limit superior equals $ 0 $’ means. Thanks!
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lim sup inequality $\limsup ( a_n b_n ) \leq \limsup a_n \limsup b_n $

I´m not sure how to start with this proof, how can I do it? $$ \limsup ( a_n b_n ) \leqslant \limsup a_n \limsup b_n $$ I also have to prove, if $ \lim a_n $ exists then: $$ \limsup ( a_n b_n ) = \limsup a_n \limsup b_n $$ Help please, it´s…
August
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Showing that two definitions of $\limsup$ are equivalent

In Rudin, $\limsup$ is defined as follows: Let $S$ be the set of subsequential limits of $\{s_n\}$. Then $$\limsup s_n = \sup S. \tag{1}$$ However, our real analysis instructor defined $\limsup$ in a different manner: $$\limsup s_n = \lim_{n…
MathMajor
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Inequality involving $\limsup$ and $\liminf$: $ \liminf(a_{n+1}/a_n) \le \liminf((a_n)^{(1/n)}) \le \limsup((a_n)^{(1/n)}) \le \limsup(a_{n+1}/a_n)$

This may have been asked before, however I was unable to find any duplicate. This comes from pg. 52 of "Mathematical Analysis: An Introduction" by Browder. Problem 14: If $(a_n)$ is a sequence in $\mathbb R$ and $a_n > 0$ for every $n$. Then show: …
podiki
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