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







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












$begingroup$


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}






share|cite|improve this answer









$endgroup$













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











Your Answer





StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3015877%2fsequence-a-n-such-that-inf-a-n-lim-inf-a-n-lim-sup-a-n-sup-a-n%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























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


















draft saved

draft discarded




















































Thanks for contributing an answer to Mathematics Stack Exchange!


  • Please be sure to answer the question. Provide details and share your research!

But avoid …



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


Use MathJax to format equations. MathJax reference.


To learn more, see our tips on writing great answers.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3015877%2fsequence-a-n-such-that-inf-a-n-lim-inf-a-n-lim-sup-a-n-sup-a-n%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown





















































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown

































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown







-

Popular posts from this blog

How to change which sound is reproduced for terminal bell?

Image
3 Recently upgrade to 18.10 from 18.04, and using the terminal is becoming annoying: the new alert sound is a drip ( /usr/share/sounds/gnome/default/alerts/drip.ogg ), and the previous one is still located at ( /usr/share/sounds/ubuntu/stereo/bell.ogg ). My issue is that at the gnome-terminal, when autocompleting or pressing left BOTH sounds are reproduced, at the same time. I want to just have ubuntu/stereo/bell.ogg active, not both . At the moment I have been only able to reduce the volume of the alerts (but still both are played), or turn the alerts down (but I dislike both "solutions"). gnome-terminal 18.10 teminal-bell share | improve this question
Read more

Can I use Tabulator js library in my java Spring + Thymeleaf project?

Image
0 I want to add Tabulator js to my java project. I added tabulator.min.js, tabulator.min.css to resources folder. After that added to my html page these dependencies as recommending in readme.md. <link rel="stylesheet" th:href="@{/css/tabulator.min.css}"> <script type="text/javascript" th:src="@{/js/tabulator.min.js}"></script> Now my tables view all elements, that I have in the html page, but not correct. In the documentation of Tabulator is example how to create header and table, var table = new Tabulator("#example-table", { columns:[ {title:"Name", field:"name", sorter:"string", width:200, editor:true}, {title:"Age", field:"age", sorter:"
Read more

Title Spacing in Bjornstrup Chapter, Removing Chapter Number From Contents

Image
up vote 5 down vote favorite 2 I would like to make it so that the vertical space below the title of each chapter is the same when using the Bjornstrup package (i.e. so that the descenders do not affect the spacing - as shown by the red arrow in the picture below). I think I'm fairly happy with the spacing around "A chapter", so I'd really like titles without descenders (e.g. "Contents") to have the same vertical spacing as "A chapter" below the title. I would also like to remove the large "0" in the contents title. Any help appreciated, as always. documentclass[11pt,a4paper]{book} % Packages[![enter image description here][1]][1] usepackage[left=3.5cm, right=3.5cm, top=4.3cm, bottom=4.25cm]{geometry} usepackage[utf8]{inputenc} usepackage{libertine} usepackage[libertine]{newtx
Read more