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
0
votes
0 answers

Limit superior and inferior, giving an example to show that it is possible to have ($a_n$) $>$ $L$ for infinitely many $n$

Question: Suppose that ($a_n$) is a sequence such that L = lim sup ($a_n$) is a real number. Then we know that, for any number $M$ $>$ $L$, there are only finitely many integers $n$ for which ($a_n$) $>$ $M$. Show by means of an example that it is…
Joy Yin
  • 129
0
votes
2 answers

Can we always find the infimum and supremum of a sequence?

I've found in my book that: $$\liminf_{n\to\infty} \ x_{n} = \sup\{\inf\{x_{k}:k\geq n \}:n \in \mathbb{N}\}$$ $$\limsup_{n\to\infty} \ x_{n} = \inf\{\sup\{x_{k}:k\geq n \}:n \in \mathbb{N}\}$$ If $X_n=\{x_{k}:k\geq n \}$ we define $s_n=\inf X_n$. I…
Jelly Belly
  • 303
0
votes
2 answers

Finding the $\liminf_{n\to\infty} \ f_n$ where $f_n = \mathbb{1}_{[n,n+1]}$?

I am trying to find: $$\liminf_{n\to\infty} \ f_n$$ where $f_n = \mathbb{1}_{[n,n+1]}$ is the indicator function that takes value $1$ in the set $[n,n+1]$ and $0$ elsewhere. It seems intuitively obvious to me that the liminf should be zero. However,…
RustyStatistician
  • 300
0
votes
1 answer

How to find the limit superior of $(-1)^n + \frac{1}{n}$?

If I have a sequence defined as $a_n = (-1)^n + \frac{1}{n}$, how would I find the limit superior?
John Sawatzky
  • 703
0
votes
1 answer

Find the limit superior of an enumeration of an interval in the rational numbers

How would I go about finding lim sup $a_n$, where $\{a_n\}$ is an enumeration of the rationals in the interval $[0,1]$? Honestly, I don't even know where to start.
John Sawatzky
  • 703
0
votes
2 answers

confusing definition of lim sup

My textbook defines: \begin{equation} \limsup (a_n) = \min\{M ∈ R |\ \ ∃n_0 \ \ ∀n > n_0, a_n ≤ M\}. \end{equation} And it gives an example: Let $a_n$= 1+ $\frac{1}{n}$. Then $\limsup (a_n)$ = 1. This confuses me - under no circumstances 1 is…
blz
  • 615
-1
votes
1 answer

Show, using the definitions, that $\lim_{n \rightarrow \infty} \inf a_n = \infty$ implies that $\lim_{n \rightarrow \infty} a_n = \infty$

Show, using the definitions, that $\lim_{n \rightarrow \infty} \inf a_n = \infty$ implies that $\lim_{n \rightarrow \infty} a_n = \infty$ Here is the definition of $\liminf$ is $\liminf_{n \rightarrow \infty} a_n = \lim(\inf{a_k : k \geq n})$…
Megan
  • 15
-1
votes
1 answer

Definition of $\limsup_{x\to 0^+}f(x)$

By definition of limit superior, we have $\limsup_{n\to \infty}x_n=\lim_{n\to \infty}(\sup_{m\geq n}x_m)$. Whats is the definition of $\limsup_{x\to 0^+}f(x)$?
M.Ramana
  • 2,753
-1
votes
1 answer

Let $U \subset\mathbb{R}^n$ and $u:U\to\mathbb{R}$ be an arbitrary function.

Let $U \subset\mathbb{R}^n$ and $u:U\to\mathbb{R}$ be an arbitrary function. Is $\lim_{r\to 0} \left(\sup_{y\in B_r(x)}u(y)\right) = \limsup\limits_{y\to x}u(y)$ always correct? "$\geq$" should always hold as long as I am not mistaken.
HelloEveryone
  • 449
-1
votes
1 answer

Prove the root test for sequence convergence with $\limsup$.

I am trying to prove the root test for sequence convergence with $\limsup$. I know exactly what to do but I think I misunderstood a simple detail when I learned about $\limsup$ and $\liminf$. That if $(a_n)$ is a sequence then we can find $N$ that…
FAF
  • 403
1 2 3 4
5