Sequence $a_n$ such that $inf a_n < lim inf a_n < lim sup a_n < sup a_n$
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
add a comment |
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
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.
– Karl
Nov 27 '18 at 15:14
add a comment |
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
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
edited Nov 27 '18 at 16:18
amWhy
192k28225439
192k28225439
asked Nov 27 '18 at 15:07
Henning WacheHenning Wache
61
61
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.
– Karl
Nov 27 '18 at 15:14
add a comment |
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.
– Karl
Nov 27 '18 at 15:14
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.
– Karl
Nov 27 '18 at 15:14
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.
– Karl
Nov 27 '18 at 15:14
add a comment |
1 Answer
1
active
oldest
votes
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}
Ok thx, this is very helpful
– Henning Wache
Nov 27 '18 at 16:14
add a comment |
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
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
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
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}
Ok thx, this is very helpful
– Henning Wache
Nov 27 '18 at 16:14
add a comment |
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}
Ok thx, this is very helpful
– Henning Wache
Nov 27 '18 at 16:14
add a comment |
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}
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}
answered Nov 27 '18 at 16:06
M.GM.G
2,2131134
2,2131134
Ok thx, this is very helpful
– Henning Wache
Nov 27 '18 at 16:14
add a comment |
Ok thx, this is very helpful
– Henning Wache
Nov 27 '18 at 16:14
Ok thx, this is very helpful
– Henning Wache
Nov 27 '18 at 16:14
Ok thx, this is very helpful
– Henning Wache
Nov 27 '18 at 16:14
add a comment |
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
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
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
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
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.
– Karl
Nov 27 '18 at 15:14