for bounded series $x_n$, prove that $limsup($1/x_n$) = 1/ liminf( $x_n$) $











up vote
0
down vote

favorite












stuck on a proof about series:
i know that $ 0 < a ≤$ $x_n$ $ ≤ b $,



a and b are real numbers (not infinite)



and from this it follows that $1/x_n$ is also bounded by 1/b and 1/a are positive



how can i prove that $ limsup( 1/ $x_n$ ) = 1 / liminf($x_n$) $



i know that $limsup(x_n) = -liminf(- $x_n$) $



and by bolzano-weirstraaus every bounded set has a converging subset..
but i cant seem to think of a solution to this one.



thanks in advance!






calculus sequences-and-series supremum-and-infimum limsup-and-liminf







share|cite|improve this question






















  • What has this to do with series? There are no series in the statement of the problem.
    – José Carlos Santos
    2 days ago










  • This is true if and only if ${x_n}$ converges.
    – Yiorgos S. Smyrlis
    2 days ago










  • Nevermind, the question is a duplicate of this one: math.stackexchange.com/questions/1164104/… Thanks anyway
    – Boaz Yakubov
    yesterday















up vote
0
down vote

favorite












stuck on a proof about series:
i know that $ 0 < a ≤$ $x_n$ $ ≤ b $,



a and b are real numbers (not infinite)



and from this it follows that $1/x_n$ is also bounded by 1/b and 1/a are positive



how can i prove that $ limsup( 1/ $x_n$ ) = 1 / liminf($x_n$) $



i know that $limsup(x_n) = -liminf(- $x_n$) $



and by bolzano-weirstraaus every bounded set has a converging subset..
but i cant seem to think of a solution to this one.



thanks in advance!






calculus sequences-and-series supremum-and-infimum limsup-and-liminf







share|cite|improve this question






















  • What has this to do with series? There are no series in the statement of the problem.
    – José Carlos Santos
    2 days ago










  • This is true if and only if ${x_n}$ converges.
    – Yiorgos S. Smyrlis
    2 days ago










  • Nevermind, the question is a duplicate of this one: math.stackexchange.com/questions/1164104/… Thanks anyway
    – Boaz Yakubov
    yesterday













up vote
0
down vote

favorite









up vote
0
down vote

favorite











stuck on a proof about series:
i know that $ 0 < a ≤$ $x_n$ $ ≤ b $,



a and b are real numbers (not infinite)



and from this it follows that $1/x_n$ is also bounded by 1/b and 1/a are positive



how can i prove that $ limsup( 1/ $x_n$ ) = 1 / liminf($x_n$) $



i know that $limsup(x_n) = -liminf(- $x_n$) $



and by bolzano-weirstraaus every bounded set has a converging subset..
but i cant seem to think of a solution to this one.



thanks in advance!






calculus sequences-and-series supremum-and-infimum limsup-and-liminf







share|cite|improve this question













stuck on a proof about series:
i know that $ 0 < a ≤$ $x_n$ $ ≤ b $,



a and b are real numbers (not infinite)



and from this it follows that $1/x_n$ is also bounded by 1/b and 1/a are positive



how can i prove that $ limsup( 1/ $x_n$ ) = 1 / liminf($x_n$) $



i know that $limsup(x_n) = -liminf(- $x_n$) $



and by bolzano-weirstraaus every bounded set has a converging subset..
but i cant seem to think of a solution to this one.



thanks in advance!






calculus sequences-and-series supremum-and-infimum limsup-and-liminf




calculus sequences-and-series supremum-and-infimum limsup-and-liminf






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked 2 days ago









Boaz Yakubov

204




204












  • What has this to do with series? There are no series in the statement of the problem.
    – José Carlos Santos
    2 days ago










  • This is true if and only if ${x_n}$ converges.
    – Yiorgos S. Smyrlis
    2 days ago










  • Nevermind, the question is a duplicate of this one: math.stackexchange.com/questions/1164104/… Thanks anyway
    – Boaz Yakubov
    yesterday


















  • What has this to do with series? There are no series in the statement of the problem.
    – José Carlos Santos
    2 days ago










  • This is true if and only if ${x_n}$ converges.
    – Yiorgos S. Smyrlis
    2 days ago










  • Nevermind, the question is a duplicate of this one: math.stackexchange.com/questions/1164104/… Thanks anyway
    – Boaz Yakubov
    yesterday
















What has this to do with series? There are no series in the statement of the problem.
– José Carlos Santos
2 days ago




What has this to do with series? There are no series in the statement of the problem.
– José Carlos Santos
2 days ago












This is true if and only if ${x_n}$ converges.
– Yiorgos S. Smyrlis
2 days ago




This is true if and only if ${x_n}$ converges.
– Yiorgos S. Smyrlis
2 days ago












Nevermind, the question is a duplicate of this one: math.stackexchange.com/questions/1164104/… Thanks anyway
– Boaz Yakubov
yesterday




Nevermind, the question is a duplicate of this one: math.stackexchange.com/questions/1164104/… Thanks anyway
– Boaz Yakubov
yesterday















active

oldest

votes











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',
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%2f3005351%2ffor-bounded-series-x-n-prove-that-limsup1-x-n-1-liminf-x-n%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown






























active

oldest

votes













active

oldest

votes









active

oldest

votes






active

oldest

votes
















 

draft saved


draft discarded



















































 


draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3005351%2ffor-bounded-series-x-n-prove-that-limsup1-x-n-1-liminf-x-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

To store a contact into the json file from server.js file using a class in NodeJS

Image
0 I have created three files -contactPresentation.js, contactService.js, contacts.json. I will run the contactService.js file. And it will navigate to contactPresentation.js file that will only choose an option to either add or display a file. If add(), then add the data into add of service file.And the data will be added to the contacts.json file . And if display(), then display the display() of service file. And the display will take the data from contacts.json file. However, i know my code is flawed. Please correct it . I am also getting an error - this.contacts.push(new ContactService(userName, contactNumber, emailId)); ^ TypeError: Cannot read property 'push' of undefined at ContactService.add (e:NodeJSProjectcontactManager2contactService.js:57:23) at new C
Read more

Redirect URL with Chrome Remote Debugging Android Devices

Image
1 How can i redirect an URL with Chrome Remote Debugging Android Devices? like Requestly Chrome extension on desktop. google-chrome debugging url redirect testing share | improve this question edited Nov 24 '18 at 8:16 sachinjain024 14.7k 23 73 129 asked Nov 23 '18 at 16:09 Momen Momen 91 10
Read more

Dieringhausen

Image
Dieringhausen Stadt Gummersbach 50.982777777778 7.5316666666667 165 Koordinaten: 50° 58′ 58″  N , 7° 31′ 54″  O Höhe: 165  (160–245)  m Einwohner: 5055  (30. Jun. 2016) Postleitzahl: 51645 Vorwahl: 02261 Lage von Dieringhausen in Gummersbach Im Hohl ca. 1927–1930 Ansicht von Südwesten bei Bünghausen Dieringhausen (im örtlichen Dialekt Dierkesen ) ist ein Ortsteil von Gummersbach im Oberbergischen Kreis, Regierungsbezirk Köln, Nordrhein-Westfalen (Deutschland). Inhaltsverzeichnis 1 Geographie 2 Geschichte 3 Kultur 3.1 Sehenswürdigkeiten 3.2 Regelmäßige Veranstaltungen 4 Infrastruktur und Wirtschaft 4.1 Verkehr 4.1.1 Schienen- und Busverkehr 4.1.2 Straßen 4.2 Öffentliche Einrichtungen 4.2.1 Schulen und Bildungseinrichtungen 4.2.2 Kirchliche Einrichtungen 5 Persönlichkeiten 5.1 In Dieringhausen geboren 5.2 In Dieringhausen gelebt 6 Einzelnachweise 7
Read more