Sequence $a_n$ such that $inf a_n < lim inf a_n < lim sup a_n < sup a_n$












1












$begingroup$


I'm searching for a sequence with the following property:



$$inf a_n < lim inf a_n < lim sup a_n < sup a_n$$



I am looking for just one example, but I am not sure how to find one.






sequences-and-series supremum-and-infimum







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  • $begingroup$
    Just take the first two terms so that they are inf and sup, for the remaining ones you take a simple alternating sequence squeezed between them.
    $endgroup$
    – Karl
    Nov 27 '18 at 15:14
















1












$begingroup$


I'm searching for a sequence with the following property:



$$inf a_n < lim inf a_n < lim sup a_n < sup a_n$$



I am looking for just one example, but I am not sure how to find one.






sequences-and-series supremum-and-infimum







share|cite|improve this question











$endgroup$












  • $begingroup$
    Just take the first two terms so that they are inf and sup, for the remaining ones you take a simple alternating sequence squeezed between them.
    $endgroup$
    – Karl
    Nov 27 '18 at 15:14














1












1








1


1



$begingroup$


I'm searching for a sequence with the following property:



$$inf a_n < lim inf a_n < lim sup a_n < sup a_n$$



I am looking for just one example, but I am not sure how to find one.






sequences-and-series supremum-and-infimum







share|cite|improve this question











$endgroup$




I'm searching for a sequence with the following property:



$$inf a_n < lim inf a_n < lim sup a_n < sup a_n$$



I am looking for just one example, but I am not sure how to find one.






sequences-and-series supremum-and-infimum




sequences-and-series supremum-and-infimum






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 27 '18 at 16:18









amWhy

1




1










asked Nov 27 '18 at 15:07









Henning WacheHenning Wache

61




61












  • $begingroup$
    Just take the first two terms so that they are inf and sup, for the remaining ones you take a simple alternating sequence squeezed between them.
    $endgroup$
    – Karl
    Nov 27 '18 at 15:14


















  • $begingroup$
    Just take the first two terms so that they are inf and sup, for the remaining ones you take a simple alternating sequence squeezed between them.
    $endgroup$
    – Karl
    Nov 27 '18 at 15:14
















$begingroup$
Just take the first two terms so that they are inf and sup, for the remaining ones you take a simple alternating sequence squeezed between them.
$endgroup$
– Karl
Nov 27 '18 at 15:14




$begingroup$
Just take the first two terms so that they are inf and sup, for the remaining ones you take a simple alternating sequence squeezed between them.
$endgroup$
– Karl
Nov 27 '18 at 15:14










1 Answer
1






active

oldest

votes


















2












$begingroup$

Here is an easy example :
begin{equation}begin{aligned} a_1 &:= 1 \ a_2 &:= 4 \ a_{2n+1} &:= 2,~~~~~text{for all $n geq 1$} \ a_{2n} &:= 3,~~~~~text{for all $n geq 1$}end{aligned}end{equation}



You can easily verify that
begin{equation}begin{aligned} inf a_n &= 1 \ liminf a_n &=2 \ limsup a_n &= 3 \ sup a_n &=4.end{aligned}end{equation}






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  • $begingroup$
    Ok thx, this is very helpful
    $endgroup$
    – Henning Wache
    Nov 27 '18 at 16:14











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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









2












$begingroup$

Here is an easy example :
begin{equation}begin{aligned} a_1 &:= 1 \ a_2 &:= 4 \ a_{2n+1} &:= 2,~~~~~text{for all $n geq 1$} \ a_{2n} &:= 3,~~~~~text{for all $n geq 1$}end{aligned}end{equation}



You can easily verify that
begin{equation}begin{aligned} inf a_n &= 1 \ liminf a_n &=2 \ limsup a_n &= 3 \ sup a_n &=4.end{aligned}end{equation}






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Ok thx, this is very helpful
    $endgroup$
    – Henning Wache
    Nov 27 '18 at 16:14
















2












$begingroup$

Here is an easy example :
begin{equation}begin{aligned} a_1 &:= 1 \ a_2 &:= 4 \ a_{2n+1} &:= 2,~~~~~text{for all $n geq 1$} \ a_{2n} &:= 3,~~~~~text{for all $n geq 1$}end{aligned}end{equation}



You can easily verify that
begin{equation}begin{aligned} inf a_n &= 1 \ liminf a_n &=2 \ limsup a_n &= 3 \ sup a_n &=4.end{aligned}end{equation}






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Ok thx, this is very helpful
    $endgroup$
    – Henning Wache
    Nov 27 '18 at 16:14














2












2








2





$begingroup$

Here is an easy example :
begin{equation}begin{aligned} a_1 &:= 1 \ a_2 &:= 4 \ a_{2n+1} &:= 2,~~~~~text{for all $n geq 1$} \ a_{2n} &:= 3,~~~~~text{for all $n geq 1$}end{aligned}end{equation}



You can easily verify that
begin{equation}begin{aligned} inf a_n &= 1 \ liminf a_n &=2 \ limsup a_n &= 3 \ sup a_n &=4.end{aligned}end{equation}






share|cite|improve this answer









$endgroup$



Here is an easy example :
begin{equation}begin{aligned} a_1 &:= 1 \ a_2 &:= 4 \ a_{2n+1} &:= 2,~~~~~text{for all $n geq 1$} \ a_{2n} &:= 3,~~~~~text{for all $n geq 1$}end{aligned}end{equation}



You can easily verify that
begin{equation}begin{aligned} inf a_n &= 1 \ liminf a_n &=2 \ limsup a_n &= 3 \ sup a_n &=4.end{aligned}end{equation}







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Nov 27 '18 at 16:06









M.GM.G

2,2281134




2,2281134












  • $begingroup$
    Ok thx, this is very helpful
    $endgroup$
    – Henning Wache
    Nov 27 '18 at 16:14


















  • $begingroup$
    Ok thx, this is very helpful
    $endgroup$
    – Henning Wache
    Nov 27 '18 at 16:14
















$begingroup$
Ok thx, this is very helpful
$endgroup$
– Henning Wache
Nov 27 '18 at 16:14




$begingroup$
Ok thx, this is very helpful
$endgroup$
– Henning Wache
Nov 27 '18 at 16:14


















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