Cauchy Sequence Characterized only By Directly Neighbouring Sequence Members Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Series constructed from a cauchy sequenceRelations among notions of convergenceCauchy Sequence proof with boundsProof review - (lack of rigour?) Convergent sequence iff Cauchy without Bolzano-WeierstrassProof verification regarding whether a certain property of a sequence implies that it is Cauchy.Why is the sequence $x(n) = log n$ **not** Cauchy?Mathematical Analysis Cauchy SequenceThat a sequence is Cauchy implies it's bounded.Determine if this specific sequence is a Cauchy sequenceCauchy sequence and boundedness

How can players take actions together that are impossible otherwise?

Can a non-EU citizen traveling with me come with me through the EU passport line?

Strange behaviour of Check

Can I throw a longsword at someone?

Writing Thesis: Copying from published papers

Why is "Captain Marvel" translated as male in Portugal?

Windows 10: How to Lock (not sleep) laptop on lid close?

Area of a 2D convex hull

Estimated State payment too big --> money back; + 2018 Tax Reform

The following signatures were invalid: EXPKEYSIG 1397BC53640DB551

How to rotate it perfectly?

Simulating Exploding Dice

What's the point in a preamp?

Can a monk deflect thrown melee weapons?

Stars Make Stars

New Order #5: where Fibonacci and Beatty meet at Wythoff

What is the largest species of polychaete?

What do you call the holes in a flute?

Working around an AWS network ACL rule limit

How do you clear the ApexPages.getMessages() collection in a test?

What was the last x86 CPU that did not have the x87 floating-point unit built in?

Complexity of many constant time steps with occasional logarithmic steps

Is there a service that would inform me whenever a new direct route is scheduled from a given airport?

Is above average number of years spent on PhD considered a red flag in future academia or industry positions?



Cauchy Sequence Characterized only By Directly Neighbouring Sequence Members



Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Series constructed from a cauchy sequenceRelations among notions of convergenceCauchy Sequence proof with boundsProof review - (lack of rigour?) Convergent sequence iff Cauchy without Bolzano-WeierstrassProof verification regarding whether a certain property of a sequence implies that it is Cauchy.Why is the sequence $x(n) = log n$ **not** Cauchy?Mathematical Analysis Cauchy SequenceThat a sequence is Cauchy implies it's bounded.Determine if this specific sequence is a Cauchy sequenceCauchy sequence and boundedness










1












$begingroup$


Let $(a_n)$ be a sequence of real numbers, for which it holds, that
$$ lim_n rightarrow infty lvert a_n+1-a_n rvert = 0. $$ Does this already imply, that $(a_n)$ is a Cauchy sequence?










share|cite|improve this question









$endgroup$
















    1












    $begingroup$


    Let $(a_n)$ be a sequence of real numbers, for which it holds, that
    $$ lim_n rightarrow infty lvert a_n+1-a_n rvert = 0. $$ Does this already imply, that $(a_n)$ is a Cauchy sequence?










    share|cite|improve this question









    $endgroup$














      1












      1








      1





      $begingroup$


      Let $(a_n)$ be a sequence of real numbers, for which it holds, that
      $$ lim_n rightarrow infty lvert a_n+1-a_n rvert = 0. $$ Does this already imply, that $(a_n)$ is a Cauchy sequence?










      share|cite|improve this question









      $endgroup$




      Let $(a_n)$ be a sequence of real numbers, for which it holds, that
      $$ lim_n rightarrow infty lvert a_n+1-a_n rvert = 0. $$ Does this already imply, that $(a_n)$ is a Cauchy sequence?







      limits cauchy-sequences






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked 3 hours ago









      Joker123Joker123

      632313




      632313




















          3 Answers
          3






          active

          oldest

          votes


















          2












          $begingroup$

          Unfortunately not. Consider
          $$a_n:=sum_i=1^nfrac1i.$$
          We find $a_n+1-a_n=1/(n+1)to 0,$ but $lim_ntoinftya_n=infty,$ hence $a_n_ninmathbbN$ is not a cauchy sequence.






          share|cite|improve this answer











          $endgroup$




















            2












            $begingroup$

            No. The sequence $a_n=sum_k=1^nfrac1k$ is a counterexample.






            share|cite|improve this answer









            $endgroup$




















              2












              $begingroup$

              Counterexample: $a_n = sqrtn$. Clearly this sequence does not converge. But
              $$
              a_n+1 - a_n = sqrtn+1 - sqrtn = frac(sqrtn+1 - sqrtn)(sqrtn+1 + sqrtn)(sqrtn+1 + sqrtn) = frac1sqrtn+1 + sqrtn to 0 , .
              $$






              share|cite|improve this answer









              $endgroup$













                Your Answer








                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%2f3188087%2fcauchy-sequence-characterized-only-by-directly-neighbouring-sequence-members%23new-answer', 'question_page');

                );

                Post as a guest















                Required, but never shown

























                3 Answers
                3






                active

                oldest

                votes








                3 Answers
                3






                active

                oldest

                votes









                active

                oldest

                votes






                active

                oldest

                votes









                2












                $begingroup$

                Unfortunately not. Consider
                $$a_n:=sum_i=1^nfrac1i.$$
                We find $a_n+1-a_n=1/(n+1)to 0,$ but $lim_ntoinftya_n=infty,$ hence $a_n_ninmathbbN$ is not a cauchy sequence.






                share|cite|improve this answer











                $endgroup$

















                  2












                  $begingroup$

                  Unfortunately not. Consider
                  $$a_n:=sum_i=1^nfrac1i.$$
                  We find $a_n+1-a_n=1/(n+1)to 0,$ but $lim_ntoinftya_n=infty,$ hence $a_n_ninmathbbN$ is not a cauchy sequence.






                  share|cite|improve this answer











                  $endgroup$















                    2












                    2








                    2





                    $begingroup$

                    Unfortunately not. Consider
                    $$a_n:=sum_i=1^nfrac1i.$$
                    We find $a_n+1-a_n=1/(n+1)to 0,$ but $lim_ntoinftya_n=infty,$ hence $a_n_ninmathbbN$ is not a cauchy sequence.






                    share|cite|improve this answer











                    $endgroup$



                    Unfortunately not. Consider
                    $$a_n:=sum_i=1^nfrac1i.$$
                    We find $a_n+1-a_n=1/(n+1)to 0,$ but $lim_ntoinftya_n=infty,$ hence $a_n_ninmathbbN$ is not a cauchy sequence.







                    share|cite|improve this answer














                    share|cite|improve this answer



                    share|cite|improve this answer








                    edited 3 hours ago









                    HAMIDINE SOUMARE

                    2,208214




                    2,208214










                    answered 3 hours ago









                    MelodyMelody

                    1,27012




                    1,27012





















                        2












                        $begingroup$

                        No. The sequence $a_n=sum_k=1^nfrac1k$ is a counterexample.






                        share|cite|improve this answer









                        $endgroup$

















                          2












                          $begingroup$

                          No. The sequence $a_n=sum_k=1^nfrac1k$ is a counterexample.






                          share|cite|improve this answer









                          $endgroup$















                            2












                            2








                            2





                            $begingroup$

                            No. The sequence $a_n=sum_k=1^nfrac1k$ is a counterexample.






                            share|cite|improve this answer









                            $endgroup$



                            No. The sequence $a_n=sum_k=1^nfrac1k$ is a counterexample.







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered 3 hours ago









                            MarkMark

                            10.6k1622




                            10.6k1622





















                                2












                                $begingroup$

                                Counterexample: $a_n = sqrtn$. Clearly this sequence does not converge. But
                                $$
                                a_n+1 - a_n = sqrtn+1 - sqrtn = frac(sqrtn+1 - sqrtn)(sqrtn+1 + sqrtn)(sqrtn+1 + sqrtn) = frac1sqrtn+1 + sqrtn to 0 , .
                                $$






                                share|cite|improve this answer









                                $endgroup$

















                                  2












                                  $begingroup$

                                  Counterexample: $a_n = sqrtn$. Clearly this sequence does not converge. But
                                  $$
                                  a_n+1 - a_n = sqrtn+1 - sqrtn = frac(sqrtn+1 - sqrtn)(sqrtn+1 + sqrtn)(sqrtn+1 + sqrtn) = frac1sqrtn+1 + sqrtn to 0 , .
                                  $$






                                  share|cite|improve this answer









                                  $endgroup$















                                    2












                                    2








                                    2





                                    $begingroup$

                                    Counterexample: $a_n = sqrtn$. Clearly this sequence does not converge. But
                                    $$
                                    a_n+1 - a_n = sqrtn+1 - sqrtn = frac(sqrtn+1 - sqrtn)(sqrtn+1 + sqrtn)(sqrtn+1 + sqrtn) = frac1sqrtn+1 + sqrtn to 0 , .
                                    $$






                                    share|cite|improve this answer









                                    $endgroup$



                                    Counterexample: $a_n = sqrtn$. Clearly this sequence does not converge. But
                                    $$
                                    a_n+1 - a_n = sqrtn+1 - sqrtn = frac(sqrtn+1 - sqrtn)(sqrtn+1 + sqrtn)(sqrtn+1 + sqrtn) = frac1sqrtn+1 + sqrtn to 0 , .
                                    $$







                                    share|cite|improve this answer












                                    share|cite|improve this answer



                                    share|cite|improve this answer










                                    answered 3 hours ago









                                    Hans EnglerHans Engler

                                    10.7k11836




                                    10.7k11836



























                                        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%2f3188087%2fcauchy-sequence-characterized-only-by-directly-neighbouring-sequence-members%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

                                        Are there any AGPL-style licences that require source code modifications to be public? Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern) Announcing the arrival of Valued Associate #679: Cesar Manara Unicorn Meta Zoo #1: Why another podcast?Force derivative works to be publicAre there any GPL like licenses for Apple App Store?Do you violate the GPL if you provide source code that cannot be compiled?GPL - is it distribution to use libraries in an appliance loaned to customers?Distributing App for free which uses GPL'ed codeModifications of server software under GPL, with web/CLI interfaceDoes using an AGPLv3-licensed library prevent me from dual-licensing my own source code?Can I publish only select code under GPLv3 from a private project?Is there published precedent regarding the scope of covered work that uses AGPL software?If MIT licensed code links to GPL licensed code what should be the license of the resulting binary program?If I use a public API endpoint that has its source code licensed under AGPL in my app, do I need to disclose my source?

                                        2013 GY136 Descoberta | Órbita | Referências Menu de navegação«List Of Centaurs and Scattered-Disk Objects»«List of Known Trans-Neptunian Objects»

                                        Mortes em março de 2019 Referências Menu de navegação«Zhores Alferov, Nobel de Física bielorrusso, morre aos 88 anos - Ciência»«Fallece Rafael Torija, o bispo emérito de Ciudad Real»«Peter Hurford dies at 88»«Keith Flint, vocalista do The Prodigy, morre aos 49 anos»«Luke Perry, ator de 'Barrados no baile' e 'Riverdale', morre aos 52 anos»«Former Rangers and Scotland captain Eric Caldow dies, aged 84»«Morreu, aos 61 anos, a antiga lenda do wrestling King Kong Bundy»«Fallece el actor y director teatral Abraham Stavans»«In Memoriam Guillaume Faye»«Sidney Sheinberg, a Force Behind Universal and Spielberg, Is Dead at 84»«Carmine Persico, Colombo Crime Family Boss, Is Dead at 85»«Dirigent Michael Gielen gestorben»«Ciclista tricampeã mundial e prata na Rio 2016 é encontrada morta em casa aos 23 anos»«Pagan Community Notes: Raven Grimassi dies, Indianapolis pop-up event cancelled, Circle Sanctuary announces new podcast, and more!»«Hal Blaine, Wrecking Crew Drummer, Dies at 90»«Morre Coutinho, que editou dupla lendária com Pelé no Santos»«Cantor Demétrius, ídolo da Jovem Guarda, morre em SP»«Ex-presidente do Vasco, Eurico Miranda morre no Rio de Janeiro»«Bronze no Mundial de basquete de 1971, Laís Elena morre aos 76 anos»«Diretor de Corridas da F1, Charlie Whiting morre aos 66 anos às vésperas do GP da Austrália»«Morreu o cardeal Danneels, da Bélgica»«Morreu o cartoonista Augusto Cid»«Morreu a atriz Maria Isabel de Lizandra, de "Vale Tudo" e novelas da Tupi»«WS Merwin, prize-winning poet of nature, dies at 91»«Atriz Márcia Real morre em São Paulo aos 88 anos»«Mauritanie: décès de l'ancien président Mohamed Mahmoud ould Louly»«Morreu Dick Dale, o rei da surf guitar e de "Pulp Fiction"»«Falleció Víctor Genes»«João Carlos Marinho, autor de 'O Gênio do Crime', morre em SP»«Legendary Horror Director and SFX Artist John Carl Buechler Dies at 66»«Morre em Salvador a religiosa Makota Valdina»«مرگ بازیکن‌ سابق نساجی بر اثر سقوط سنگ در مازندران»«Domingos Oliveira morre no Rio»«Morre Airton Ravagniani, ex-São Paulo, Fla, Vasco, Grêmio e Sport - Notícias»«Morre o escritor Flavio Moreira da Costa»«Larry Cohen, Writer-Director of 'It's Alive' and 'Hell Up in Harlem,' Dies at 77»«Scott Walker, experimental singer-songwriter, dead at 76»«Joseph Pilato, Day of the Dead Star and Horror Favorite, Dies at 70»«Sheffield United set to pay tribute to legendary goalkeeper Ted Burgin who has died at 91»«Morre Rafael Henzel, sobrevivente de acidente aéreo da Chapecoense»«Morre Valery Bykovsky, um dos primeiros cosmonautas da União Soviética»«Agnès Varda, cineasta da Nouvelle Vague, morre aos 90 anos»«Agnès Varda, cineasta francesa, morre aos 90 anos»«Tania Mallet, James Bond Actress and Helen Mirren's Cousin, Dies at 77»e