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
11
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5 answers

Understanding limsup

My textbook says: $$\overline{s}_n = \sup \{a_n \mid n \geq N\}$$ and $\operatorname{limsup} \{a_n\}_{n \to \infty} = \lim_{N \to \infty} \overline{s}_N$. Also, it says: As $N$ gets larger, the sup is taken over a smaller set, so the sequence of…
kiwifruit
  • 707
9
votes
2 answers

O-Notation: Limsup vs. Lim

This is the big O-Notation as given today in one of our exercise classes (it is sloppy but just as it was on the blackboard): $$\begin{align*} &f = O(g):\quad\limsup_{x \rightarrow a}\, \left|\frac{f}{g}\right| < \infty\\\\ &f = o(g):\quad\lim_{x…
user62487
  • 483
8
votes
1 answer

Composition of sets with positive upper density

Let $A$ be a set of positive integers $\{a_n: n\ge 1\}$ (in increasing order) with positive upper density, that is, $$ \limsup_{n\to \infty}\frac{1}{n}|A \cap [1,n]|>0. $$ Let also $B=\{b_n: n\ge 1\}$ a set of positive upper density. Is it true that…
user207096
7
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1 answer

What's the relation between $\limsup$ and $\inf\sup$?

I've found in my book that: $$\liminf_{n\to\infty} \ x_{n} = \sup\{\inf\{x_{k}:k\geq n \}:n \in \mathbb{N}\}$$ $$\limsup_{n\to\infty} \ x_{n} = \inf\{\sup\{x_{k}:k\geq n \}:n \in \mathbb{N}\}$$ But I don't understand why. According to my book the…
Jelly Belly
  • 303
2
votes
2 answers

Prove that limsup is the supremum of the limit points

Let ${a_n}$ be a bounded sequence of real numbers, and let P be the set of limit points. Prove that $\limsup a_n = \sup P$. I have proved that there must be a subsequence that converges to limsup$a_n$. But I am stuck connecting this to the maximum…
user146219
  • 23
2
votes
0 answers

Proof that the limit inferior is less than or equal to the limit superior of a sequence

I have thought of a proof that the limit inferior of a sequence is less than or equal to the limit superior of that sequence as part of Exercise 6.4.3 part (c) from Tao's book Analysis I Fourth Edition. In particular, I approached this with a proof…
Vaskara_GRek_O
  • 119
2
votes
1 answer

Prove that there exists a sequence $x_n\to\infty$ such that $\lim_{n\to\infty} f(x_n) = b$ and $\lim_{n\to\infty} f'(x_n) = 0$

Let $f(x)$ be a real-valued differentiable function on $[0,\infty)$ and $\limsup_{x\to\infty}f(x)=b>\liminf_{x\to\infty} f(x)=a$. How to prove that there exists a sequence $x_n\to\infty$ such that $\lim_{n\to\infty} f(x_n) = b$ and…
Stephen
  • 786
2
votes
1 answer

Showing upper semicontinuity of $u^*(x) = \lim\limits_{r\to 0}\left( \sup\limits_{B_r(x)}u \right)$.

Morning. I had a similar post yesterday but I noticed after a good sleep tonight that whad I did there does not work. As the post says I want to show that $u^*(x) = \lim\limits_{r\to 0}\left( \sup\limits_{B_r(x)}u \right)$ is upper semicontinuous…
HelloEveryone
  • 449
2
votes
3 answers

Prove $\liminf$ definition

This is homework assignment. Let $(x_n)$ be bounded sequence. Prove following equation $$\liminf_{n \rightarrow \infty}\, x_n = \max \{ B \in \mathbb{R} : \forall \varepsilon > 0 \{n \in \mathbb{N}: x_n \leq B - \varepsilon\} \; \text{is finite…
flowian
  • 541
2
votes
1 answer

Lim sup definition

I have a problem with the lim sup definition when it is applied to a sequence of functions. Consider the sets $(A_{n})_{n\in\mathbb{N}}$. They are all pairwise disjoint. Then we have a sequence of functions given: $f_{n} = 1_{A_{n}}$ My problem…
whoisitnow
  • 131
2
votes
1 answer

Further clarification on the connection between limsup and liminf for sequences of sets and real numbers

Big picture goal: I am trying to reconcile the difference between between the definition of $\liminf,\limsup$ of a sequence of real numbers and a sequence of sets for the special case of the discrete metric (as per wikipedia). liminf: I know that…
user79950
  • 549
2
votes
0 answers

When working with the limites inferior and superior, how do you procede?

There are typical cases of various complexity where notions are defined or solvable by liminf or limsup. At the same time, from my perception, they are touched only superficially during the course of study up until (mostly facultative) measury…
user16403
  • 41
1
vote
0 answers

limit superior and limit inferior of $\frac{n+(-1)^n n^2}{n^2+1}$

Find limit superior and limit inferior of $\frac{n+(-1)^n n^2}{n^2+1}$: the subsequential limit is $1$ and $-1$. so the limit $\sup = 1$ and limit $\inf = -1$ let $E_k = \{a_n\mid n>k\}$. $$E_k= \begin{cases}…
jason
  • 129
1
vote
1 answer

Is this limsup calculation correct?

$a_n=1$ if $n=2^k$ for $k>0$ $a_n=\frac{1}{n!}$ otherwise a) Find limsup $\displaystyle \frac{|a_{n+1}|}{|a_n|}$ - I think this is infinity because we can find a term that is 1/something!, but the next term will be 1, so the ratio will be something!…
user146214
  • 13
1
vote
1 answer

$\limsup$ of series

Find $\limsup\limits_{j\to\infty} |a_j|^{1/j}$, where $$a_j = \sum_{j=1}^\infty \frac{(1+1/j)^{2j}}{e^j}$$ Since the limit of the numerator is $e^2$, is it correct that the $\limsup$ is equal to $0$? How can I write out the formal calculation?
kiwifruit
  • 707
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