Mathematics Stack Exchange
Mathematics Stack Exchange

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.

1947 questions
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updated: lim inf, lim sup and bertrand's postulate for a function

Define $q:\mathbb{N}\rightarrow\mathbb{Q}^+$ with, $q(n)=min\{\alpha\in \mathbb{Q}^+\mid \exists \;p\in \mathbb{N},\; p \text{ prime and } p \in (n,\alpha n]\}$. Is the $\liminf\limits_{n\rightarrow\infty} q(n)=1$? I was thinking that for $p$ prime,…
snulty
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Counterexample for limsup

Statement: $\limsup\limits_{n\to\infty} c_n a_n = c \limsup\limits_{n\to\infty} a_n$ Please help find a counterexample to this statement if $c<0$. Edit: also suppose $c_n \to c$ and $\limsup a_n$ is finite
kiwifruit
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Product of lim sups

Suppose lim sup $a_n$ is finite, and $c_n \to c$ Prove that if $c \geq 0$ lim sup $a_n c_n$ = c lim sum $a_n$ and find a counterexample to this if $c <0$. Is there a rule that the product of lim sups is equal to the lim sup of the product? Also,…
kiwifruit
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$P(\liminf S_n)=1$ what does it mean?

$P(\cdot)$ is the probability measure. $S_n$ is a sequence of events. $P(\liminf S_n)=1$ does it mean that $S_n$ always happen after some large n? Can I say that it must be true that $\liminf P(S_n)=1$ given $P(\liminf S_n)=1$ ?
lulu
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Pointwise convergence and liminf

Convergence in measure is strictly weaker than pointwise convergence. To see this, take $(X, \mathcal{A} , \mu)$ = $([0, 1], \mathcal{B}[0, 1], λ|[0,1])$ and set: $u_n(x) := \textbf{1}_{[i2^{−k},(i+1)2^{−k}]}(x)$, where $n = i + 2^k, 0\leq i…
Ricter
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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 subsets $\left(B_{n}:n\in \mathbb{N}\right)$ by…
Kr'aamkh
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How do I prove that this set is a Borel set?

I have the following Problem: Let $f_n:\mathbb{R}\rightarrow \mathbb{R}$ be Borel-mesurable functions. Show that $\{x|\limsup\,f_n(x)=\infty\}$ is borel set. In the lecture, we did a similar proof with $\lim f_n(x)$ instead of $\limsup$. There we…
user123234
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Why is does proposition about upper limit set hold?

Let $X_n$ be a set and let $\displaystyle\overline\lim_{n\to \infty} X_n=\bigcap_{k=1}^{\infty} \bigcup_{n=k}^{\infty} X_n$. Prove that $ x \in \displaystyle\overline\lim_{n\to \infty} X_n \iff$ There exists $\{ n_k \}_{k=1}^{\infty} \subset…
daㅤ
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Do $\limsup$ and $\liminf$ are the actual bounds of a sequence?

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 $a_n$?
TheAsker
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Clarification on lim inf and lim sup of a serie of sets

I'm trying to understand what $\limsup A_n= \bigcap_{N=1}^\infty \left( \bigcup_{n\ge N} A_n \right)$ and $\liminf A_n = \bigcup_{N=1}^\infty \left(\bigcap_{n \ge N} A_n\right)$ actually means. I've already seen this answer, but there is something I…
MoonlessNightss
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Proving $\Theta(n^{k})= {n \choose k}$

I need help proving $n\choose k$$=\Theta(n^{k})$ where $f(n)=\Theta(g(n))$ if $0<\liminf\limits_{n\rightarrow\infty}\frac{g(n)}{f(n)}\le\limsup\limits_{n\rightarrow\infty}\frac{g(n)}{f(n)}<\infty$ Can someone show me the calculation?
Frederick
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$V_{\epsilon}(x^*)$ notation

What does the following notation $V_{\epsilon}(x^*)$ in the context and there are infinitely many terms of $a_n$ in $V_{\epsilon}(x^*)$ on this link.
Michael Munta
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sup$_{x\in S}x$ notation for lim sup/inf

What does the following notation 'sup$_{x\in S}x$' mean in the context of limit superior and inferior? I am asking because usually I see it used like 'sup$X$' for some set $X$, but here it seems as if an element is passed to it.
Michael Munta
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Separate tails of sequences in defining lim sup/inf

Why is on the following link a limit superior of a sequence $$a_n=(-1)^n/n$$ defined separately for $n$ odd and even? Namely for $n$ odd its $A_n=1/(n+1)$ and for even its $A_n=1/n$. What does it even mean in this case when two sequences of…
Michael Munta
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About some Bachmann–Landau notations

Let $(u_{n})_{n},(v_{n})_{n}$ two strictly positive real sequences verifying the following property: There exist a positive integer $N$ and there exist a positive real number a such that for all $n>N$ we have $$u_{n}/v_{n}≤a$$ I am asking if there…
Safwane
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