Cauchy Sequence Characterized only By Directly Neighbouring Sequence Members Announcing the...

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

Antler Helmet: Can it work?

Replacing HDD with SSD; what about non-APFS/APFS?

Direct Experience of Meditation

Cold is to Refrigerator as warm is to?

3 doors, three guards, one stone

Can the prologue be the backstory of your main character?

Estimate capacitor parameters

What do you call the holes in a flute?

Working around an AWS network ACL rule limit

Why don't the Weasley twins use magic outside of school if the Trace can only find the location of spells cast?

Determine whether f is a function, an injection, a surjection

Who can trigger ship-wide alerts in Star Trek?

Can I throw a longsword at someone?

Did the new image of black hole confirm the general theory of relativity?

Complexity of many constant time steps with occasional logarithmic steps

If I can make up priors, why can't I make up posteriors?

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

Stars Make Stars

What is the largest species of polychaete?

Problem when applying foreach loop

Notation for two qubit composite product state

Using "nakedly" instead of "with nothing on"

What are the performance impacts of 'functional' Rust?



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?






limits cauchy-sequences







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?






    limits cauchy-sequences







    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?






      limits cauchy-sequences







      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




      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
          $$a_n:=sum_{i=1}^nfrac{1}{i}.$$
          We find $a_{n+1}-a_n=1/(n+1)to 0,$ but $lim_{ntoinfty}a_n=infty,$ hence ${a_n}_{ninmathbb{N}}$ is not a cauchy sequence.






          share|cite|improve this answer











          $endgroup$





















            2












            $begingroup$

            No. The sequence $a_n=sum_{k=1}^nfrac{1}{k}$ is a counterexample.






            share|cite|improve this answer









            $endgroup$





















              2












              $begingroup$

              Counterexample: $a_n = sqrt{n}$. Clearly this sequence does not converge. But
              $$
              a_{n+1} - a_{n} = sqrt{n+1} - sqrt{n} = frac{(sqrt{n+1} - sqrt{n})(sqrt{n+1} + sqrt{n})}{(sqrt{n+1} + sqrt{n})} = frac{1}{sqrt{n+1} + sqrt{n}} to 0 , .
              $$






              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
                $$a_n:=sum_{i=1}^nfrac{1}{i}.$$
                We find $a_{n+1}-a_n=1/(n+1)to 0,$ but $lim_{ntoinfty}a_n=infty,$ hence ${a_n}_{ninmathbb{N}}$ is not a cauchy sequence.






                share|cite|improve this answer











                $endgroup$


















                  2












                  $begingroup$

                  Unfortunately not. Consider
                  $$a_n:=sum_{i=1}^nfrac{1}{i}.$$
                  We find $a_{n+1}-a_n=1/(n+1)to 0,$ but $lim_{ntoinfty}a_n=infty,$ hence ${a_n}_{ninmathbb{N}}$ is not a cauchy sequence.






                  share|cite|improve this answer











                  $endgroup$
















                    2












                    2








                    2





                    $begingroup$

                    Unfortunately not. Consider
                    $$a_n:=sum_{i=1}^nfrac{1}{i}.$$
                    We find $a_{n+1}-a_n=1/(n+1)to 0,$ but $lim_{ntoinfty}a_n=infty,$ hence ${a_n}_{ninmathbb{N}}$ is not a cauchy sequence.






                    share|cite|improve this answer











                    $endgroup$



                    Unfortunately not. Consider
                    $$a_n:=sum_{i=1}^nfrac{1}{i}.$$
                    We find $a_{n+1}-a_n=1/(n+1)to 0,$ but $lim_{ntoinfty}a_n=infty,$ hence ${a_n}_{ninmathbb{N}}$ is not a cauchy sequence.







                    share|cite|improve this answer














                    share|cite|improve this answer



                    share|cite|improve this answer








                    edited 2 hours ago









                    HAMIDINE SOUMARE

                    2,208214




                    2,208214










                    answered 3 hours ago









                    MelodyMelody

                    1,27012




                    1,27012























                        2












                        $begingroup$

                        No. The sequence $a_n=sum_{k=1}^nfrac{1}{k}$ is a counterexample.






                        share|cite|improve this answer









                        $endgroup$


















                          2












                          $begingroup$

                          No. The sequence $a_n=sum_{k=1}^nfrac{1}{k}$ is a counterexample.






                          share|cite|improve this answer









                          $endgroup$
















                            2












                            2








                            2





                            $begingroup$

                            No. The sequence $a_n=sum_{k=1}^nfrac{1}{k}$ is a counterexample.






                            share|cite|improve this answer









                            $endgroup$



                            No. The sequence $a_n=sum_{k=1}^nfrac{1}{k}$ 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: $a_n = sqrt{n}$. Clearly this sequence does not converge. But
                                $$
                                a_{n+1} - a_{n} = sqrt{n+1} - sqrt{n} = frac{(sqrt{n+1} - sqrt{n})(sqrt{n+1} + sqrt{n})}{(sqrt{n+1} + sqrt{n})} = frac{1}{sqrt{n+1} + sqrt{n}} to 0 , .
                                $$






                                share|cite|improve this answer









                                $endgroup$


















                                  2












                                  $begingroup$

                                  Counterexample: $a_n = sqrt{n}$. Clearly this sequence does not converge. But
                                  $$
                                  a_{n+1} - a_{n} = sqrt{n+1} - sqrt{n} = frac{(sqrt{n+1} - sqrt{n})(sqrt{n+1} + sqrt{n})}{(sqrt{n+1} + sqrt{n})} = frac{1}{sqrt{n+1} + sqrt{n}} to 0 , .
                                  $$






                                  share|cite|improve this answer









                                  $endgroup$
















                                    2












                                    2








                                    2





                                    $begingroup$

                                    Counterexample: $a_n = sqrt{n}$. Clearly this sequence does not converge. But
                                    $$
                                    a_{n+1} - a_{n} = sqrt{n+1} - sqrt{n} = frac{(sqrt{n+1} - sqrt{n})(sqrt{n+1} + sqrt{n})}{(sqrt{n+1} + sqrt{n})} = frac{1}{sqrt{n+1} + sqrt{n}} to 0 , .
                                    $$






                                    share|cite|improve this answer









                                    $endgroup$



                                    Counterexample: $a_n = sqrt{n}$. Clearly this sequence does not converge. But
                                    $$
                                    a_{n+1} - a_{n} = sqrt{n+1} - sqrt{n} = frac{(sqrt{n+1} - sqrt{n})(sqrt{n+1} + sqrt{n})}{(sqrt{n+1} + sqrt{n})} = frac{1}{sqrt{n+1} + sqrt{n}} to 0 , .
                                    $$







                                    share|cite|improve this answer












                                    share|cite|improve this answer



                                    share|cite|improve this answer










                                    answered 2 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

                                        “%fieldName is a required field.”, in Magento2 REST API Call for GET Method Type The Next...

                                        Image
                                        How seriously should I take size and weight limits of hand luggage? Can a virus destroy the BIOS of a modern computer? Can I hook these wires up to find the connection to a dead outlet? How to keep a dark protagonist who wants to keep his humanity dark? How does a dynamic QR code work? My singleton can be called multiple times How to detect integer overflow in C? Can a PhD from a non-TU9 German university become a professor in a TU9 university? How to travel Japan while expressing milk? Is it a bad idea to plug the other end of ESD strap to wall ground? How to show a landlord what we have in s
                                        Read more

                                        How to change City field to a dropdown in Checkout step Magento 2Magento 2 : How to change UI field(s)...

                                        Image
                                        Does the US political system, in principle, allow for a no-party system? Is it a Cyclops number? "Nobody" knows! Understanding a proof about a set being closed Rationale to prefer local variables over instance variables? Movie: boy escapes the real world and goes to a fantasy world with big furry trolls Why aren't there more Gauls like Obelix? I reported the illegal activity of my boss to his boss. My boss found out. Now I am being punished. What should I do? Solving Linear Matrix Recurrences What is the generally accepted pronunciation of “topoi”? Is it safe to abruplty remove arduino power
                                        Read more

                                        夢乃愛華...

                                        Image
                                        1995年出生在世人物日本AV女優S1女優埼玉縣出身人物 日本AV女優S1TeStar千葉縣松戶市立旭町中學校松戶市立松戶高等學校淑德大學S1 body.skin-minerva .mw-parser-output table.infobox caption{text-align:center} ゆめの あいか 夢乃 あいか AV女优 别名 安藤琴乃（本名） [1] 民族 大和族 出生 ( 1995-09-18 ) 1995年9月18日 （ 23歲） 出道地点   日本宮城縣 → 千葉縣 → 埼玉縣 活跃年代 2013年 - 血型 B型 体形（2013年 [2] 数值） 身高／体重 149 厘米 / ― 公斤 三围 79 - 52 - 78 cm 罩杯 G 体形备注 巨乳 计量单位换算 身高／体重 4英尺11英寸 / ― 磅 三围 31 - 20 - 31 英寸 活动 拍摄类型 AV 模特儿类型 水着 其他职业 鯛魚燒店員（求學時期） [1] 業餘攝影會模特兒（求學時期） [3] 教学：Template:AV女优/doc 夢乃愛華 （日语： 夢乃 あいか ／ ゆめの あいか   Yumeno Aika ，1995年9月18日 － [1] ）是日本AV女優，现为S1专属女优，所屬經紀公司為TeStar。 目录 1 簡歷 2 演出作品 2.1 成人影片 2.2 形象影片 2.3 綜藝節目 2.4 網上節目 3 其他產品 3.1 雜誌 3.2 寫真集 3.3 網上寫真 4 相關條目 5 參考資料 6 外部連結 簡歷 夢乃愛華出身於宮城縣，有一位姊姊 [3] ，後與家人移居千葉縣，曾就讀當地的松戶市立旭町中學校及松戶市立松戶高等學校 [1] 。她中學一年級時身高已停止增長，而高中初期胸部罩杯亦只有 B，但隨著「初體驗」和均衡飲食帶來的生理影響，三年級畢業那刻罩杯則升至 F [3] 。
                                        Read more