# limsup and liminf

• Apr 4th 2013, 05:35 AM
rokman54
limsup and liminf
I have a question on the definition of limsup and liminf.

First let $A_k = [a_n | n \geq k}$ with $A_k \subset A_{k-1} \subset ... \subset A_1$

s_n = SupA_n, l_n = Inf A_n

$s_1 \geq s_2 ... \geq s_n, l_1 \leq l_2 ... \leq l_n$

$limsup a_n = lim_{k -> \infty} Sup A$

$liminf a_n= lim_{k -> \infty} Inf A$

I don't understand $s_1 \geq s_2 ... \geq s_n, l_1 \leq l_2 ... \leq l_n$

Why is $l_1 \leq l_2 ... \leq l_n$
• Apr 4th 2013, 06:15 AM
Plato
Re: limsup and liminf
Quote:

Originally Posted by rokman54
Why is $l_1 \leq l_2 ... \leq l_n$

For bounded sets $C~\&~D$ if $C\subset D$ then surely you understand that
$\sup(D)\ge\sup(C)~\&~\inf(D)\le\inf(C)~?$

In this case $A_n\subset A_1$ so $l_1\le l_n$.