Construct a sequence subsequential limits of which form a range $[a, b]subset Bbb R$












1












$begingroup$


I want to:




construct a sequence subsequential limits of which form a range $[a, b]subset Bbb R$




Not sure how to express myself correctly but I will try.



I was thinking of a range from $a$ to $b$. If i could split that range infinitely many times and always choose either left or right part, then I could use that "path" as a subsequence. From that i should be able to find infinitely many subsequences which with some reordering will form a sequence containing any number in the range $[a, b]$ as its subsequential limit.



I've been playing around with this idea in desmos, here is a basic sketch of the idea.



Could someone help me construct such a sequence? Is my approach even valid? If so then how could i finish it?






calculus sequences-and-series limits







share|cite|improve this question











$endgroup$












  • $begingroup$
    enumerate rationals in $[a,b]$
    $endgroup$
    – mathworker21
    Dec 16 '18 at 15:48










  • $begingroup$
    I can't see anything but empty space in desmos.
    $endgroup$
    – BigbearZzz
    Dec 16 '18 at 15:52










  • $begingroup$
    @BigbearZzz I've fixed the link, thanks for pointing out
    $endgroup$
    – roman
    Dec 16 '18 at 15:53
















1












$begingroup$


I want to:




construct a sequence subsequential limits of which form a range $[a, b]subset Bbb R$




Not sure how to express myself correctly but I will try.



I was thinking of a range from $a$ to $b$. If i could split that range infinitely many times and always choose either left or right part, then I could use that "path" as a subsequence. From that i should be able to find infinitely many subsequences which with some reordering will form a sequence containing any number in the range $[a, b]$ as its subsequential limit.



I've been playing around with this idea in desmos, here is a basic sketch of the idea.



Could someone help me construct such a sequence? Is my approach even valid? If so then how could i finish it?






calculus sequences-and-series limits







share|cite|improve this question











$endgroup$












  • $begingroup$
    enumerate rationals in $[a,b]$
    $endgroup$
    – mathworker21
    Dec 16 '18 at 15:48










  • $begingroup$
    I can't see anything but empty space in desmos.
    $endgroup$
    – BigbearZzz
    Dec 16 '18 at 15:52










  • $begingroup$
    @BigbearZzz I've fixed the link, thanks for pointing out
    $endgroup$
    – roman
    Dec 16 '18 at 15:53














1












1








1





$begingroup$


I want to:




construct a sequence subsequential limits of which form a range $[a, b]subset Bbb R$




Not sure how to express myself correctly but I will try.



I was thinking of a range from $a$ to $b$. If i could split that range infinitely many times and always choose either left or right part, then I could use that "path" as a subsequence. From that i should be able to find infinitely many subsequences which with some reordering will form a sequence containing any number in the range $[a, b]$ as its subsequential limit.



I've been playing around with this idea in desmos, here is a basic sketch of the idea.



Could someone help me construct such a sequence? Is my approach even valid? If so then how could i finish it?






calculus sequences-and-series limits







share|cite|improve this question











$endgroup$




I want to:




construct a sequence subsequential limits of which form a range $[a, b]subset Bbb R$




Not sure how to express myself correctly but I will try.



I was thinking of a range from $a$ to $b$. If i could split that range infinitely many times and always choose either left or right part, then I could use that "path" as a subsequence. From that i should be able to find infinitely many subsequences which with some reordering will form a sequence containing any number in the range $[a, b]$ as its subsequential limit.



I've been playing around with this idea in desmos, here is a basic sketch of the idea.



Could someone help me construct such a sequence? Is my approach even valid? If so then how could i finish it?






calculus sequences-and-series limits




calculus sequences-and-series limits






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 16 '18 at 15:53







roman

















asked Dec 16 '18 at 15:46









romanroman

2,34321224




2,34321224












  • $begingroup$
    enumerate rationals in $[a,b]$
    $endgroup$
    – mathworker21
    Dec 16 '18 at 15:48










  • $begingroup$
    I can't see anything but empty space in desmos.
    $endgroup$
    – BigbearZzz
    Dec 16 '18 at 15:52










  • $begingroup$
    @BigbearZzz I've fixed the link, thanks for pointing out
    $endgroup$
    – roman
    Dec 16 '18 at 15:53


















  • $begingroup$
    enumerate rationals in $[a,b]$
    $endgroup$
    – mathworker21
    Dec 16 '18 at 15:48










  • $begingroup$
    I can't see anything but empty space in desmos.
    $endgroup$
    – BigbearZzz
    Dec 16 '18 at 15:52










  • $begingroup$
    @BigbearZzz I've fixed the link, thanks for pointing out
    $endgroup$
    – roman
    Dec 16 '18 at 15:53
















$begingroup$
enumerate rationals in $[a,b]$
$endgroup$
– mathworker21
Dec 16 '18 at 15:48




$begingroup$
enumerate rationals in $[a,b]$
$endgroup$
– mathworker21
Dec 16 '18 at 15:48












$begingroup$
I can't see anything but empty space in desmos.
$endgroup$
– BigbearZzz
Dec 16 '18 at 15:52




$begingroup$
I can't see anything but empty space in desmos.
$endgroup$
– BigbearZzz
Dec 16 '18 at 15:52












$begingroup$
@BigbearZzz I've fixed the link, thanks for pointing out
$endgroup$
– roman
Dec 16 '18 at 15:53




$begingroup$
@BigbearZzz I've fixed the link, thanks for pointing out
$endgroup$
– roman
Dec 16 '18 at 15:53










3 Answers
3






active

oldest

votes


















3












$begingroup$

When I click on your link, I just get the Desmos home page.



On the other hand, you can take the sequence:$$a,b,a,frac12a+frac12b,b,a,frac23a+frac13b,frac13a+frac23b,b,a,frac34a+frac14b,ldots$$






share|cite|improve this answer











$endgroup$













  • $begingroup$
    That's the one! I just could't wrap my mind around it, thank you!
    $endgroup$
    – roman
    Dec 16 '18 at 15:57






  • 1




    $begingroup$
    I just realized that my answer is essentially the same as you. I should have read carefully first.
    $endgroup$
    – BigbearZzz
    Dec 16 '18 at 15:59



















2












$begingroup$

If $[a,b]=[0,1]$ then you can take the sequence
$$
frac 11, frac 12, frac 22, frac 13, frac 23, frac 33, frac 14, frac 24, frac 34, frac 44, dots
$$
It shouldn't be hard to see that any point in $[0,1]$ can be approximated arbitrarily well.



For general $[a,b]$ just linearly transform the above.






share|cite|improve this answer









$endgroup$





















    2












    $begingroup$

    Its famous that {$sin (n)}_{n=1}^infty$ is dense in $[-1,+1]$. So, shift to the interval of interest:
    $$ {a+ (b-a) sin n}_{nin mathbb{N}} .$$






    share|cite|improve this answer









    $endgroup$













      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%2f3042746%2fconstruct-a-sequence-subsequential-limits-of-which-form-a-range-a-b-subset%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









      3












      $begingroup$

      When I click on your link, I just get the Desmos home page.



      On the other hand, you can take the sequence:$$a,b,a,frac12a+frac12b,b,a,frac23a+frac13b,frac13a+frac23b,b,a,frac34a+frac14b,ldots$$






      share|cite|improve this answer











      $endgroup$













      • $begingroup$
        That's the one! I just could't wrap my mind around it, thank you!
        $endgroup$
        – roman
        Dec 16 '18 at 15:57






      • 1




        $begingroup$
        I just realized that my answer is essentially the same as you. I should have read carefully first.
        $endgroup$
        – BigbearZzz
        Dec 16 '18 at 15:59
















      3












      $begingroup$

      When I click on your link, I just get the Desmos home page.



      On the other hand, you can take the sequence:$$a,b,a,frac12a+frac12b,b,a,frac23a+frac13b,frac13a+frac23b,b,a,frac34a+frac14b,ldots$$






      share|cite|improve this answer











      $endgroup$













      • $begingroup$
        That's the one! I just could't wrap my mind around it, thank you!
        $endgroup$
        – roman
        Dec 16 '18 at 15:57






      • 1




        $begingroup$
        I just realized that my answer is essentially the same as you. I should have read carefully first.
        $endgroup$
        – BigbearZzz
        Dec 16 '18 at 15:59














      3












      3








      3





      $begingroup$

      When I click on your link, I just get the Desmos home page.



      On the other hand, you can take the sequence:$$a,b,a,frac12a+frac12b,b,a,frac23a+frac13b,frac13a+frac23b,b,a,frac34a+frac14b,ldots$$






      share|cite|improve this answer











      $endgroup$



      When I click on your link, I just get the Desmos home page.



      On the other hand, you can take the sequence:$$a,b,a,frac12a+frac12b,b,a,frac23a+frac13b,frac13a+frac23b,b,a,frac34a+frac14b,ldots$$







      share|cite|improve this answer














      share|cite|improve this answer



      share|cite|improve this answer








      edited Dec 16 '18 at 15:57

























      answered Dec 16 '18 at 15:52









      José Carlos SantosJosé Carlos Santos

      166k22132235




      166k22132235












      • $begingroup$
        That's the one! I just could't wrap my mind around it, thank you!
        $endgroup$
        – roman
        Dec 16 '18 at 15:57






      • 1




        $begingroup$
        I just realized that my answer is essentially the same as you. I should have read carefully first.
        $endgroup$
        – BigbearZzz
        Dec 16 '18 at 15:59


















      • $begingroup$
        That's the one! I just could't wrap my mind around it, thank you!
        $endgroup$
        – roman
        Dec 16 '18 at 15:57






      • 1




        $begingroup$
        I just realized that my answer is essentially the same as you. I should have read carefully first.
        $endgroup$
        – BigbearZzz
        Dec 16 '18 at 15:59
















      $begingroup$
      That's the one! I just could't wrap my mind around it, thank you!
      $endgroup$
      – roman
      Dec 16 '18 at 15:57




      $begingroup$
      That's the one! I just could't wrap my mind around it, thank you!
      $endgroup$
      – roman
      Dec 16 '18 at 15:57




      1




      1




      $begingroup$
      I just realized that my answer is essentially the same as you. I should have read carefully first.
      $endgroup$
      – BigbearZzz
      Dec 16 '18 at 15:59




      $begingroup$
      I just realized that my answer is essentially the same as you. I should have read carefully first.
      $endgroup$
      – BigbearZzz
      Dec 16 '18 at 15:59











      2












      $begingroup$

      If $[a,b]=[0,1]$ then you can take the sequence
      $$
      frac 11, frac 12, frac 22, frac 13, frac 23, frac 33, frac 14, frac 24, frac 34, frac 44, dots
      $$
      It shouldn't be hard to see that any point in $[0,1]$ can be approximated arbitrarily well.



      For general $[a,b]$ just linearly transform the above.






      share|cite|improve this answer









      $endgroup$


















        2












        $begingroup$

        If $[a,b]=[0,1]$ then you can take the sequence
        $$
        frac 11, frac 12, frac 22, frac 13, frac 23, frac 33, frac 14, frac 24, frac 34, frac 44, dots
        $$
        It shouldn't be hard to see that any point in $[0,1]$ can be approximated arbitrarily well.



        For general $[a,b]$ just linearly transform the above.






        share|cite|improve this answer









        $endgroup$
















          2












          2








          2





          $begingroup$

          If $[a,b]=[0,1]$ then you can take the sequence
          $$
          frac 11, frac 12, frac 22, frac 13, frac 23, frac 33, frac 14, frac 24, frac 34, frac 44, dots
          $$
          It shouldn't be hard to see that any point in $[0,1]$ can be approximated arbitrarily well.



          For general $[a,b]$ just linearly transform the above.






          share|cite|improve this answer









          $endgroup$



          If $[a,b]=[0,1]$ then you can take the sequence
          $$
          frac 11, frac 12, frac 22, frac 13, frac 23, frac 33, frac 14, frac 24, frac 34, frac 44, dots
          $$
          It shouldn't be hard to see that any point in $[0,1]$ can be approximated arbitrarily well.



          For general $[a,b]$ just linearly transform the above.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 16 '18 at 15:58









          BigbearZzzBigbearZzz

          8,92021652




          8,92021652























              2












              $begingroup$

              Its famous that {$sin (n)}_{n=1}^infty$ is dense in $[-1,+1]$. So, shift to the interval of interest:
              $$ {a+ (b-a) sin n}_{nin mathbb{N}} .$$






              share|cite|improve this answer









              $endgroup$


















                2












                $begingroup$

                Its famous that {$sin (n)}_{n=1}^infty$ is dense in $[-1,+1]$. So, shift to the interval of interest:
                $$ {a+ (b-a) sin n}_{nin mathbb{N}} .$$






                share|cite|improve this answer









                $endgroup$
















                  2












                  2








                  2





                  $begingroup$

                  Its famous that {$sin (n)}_{n=1}^infty$ is dense in $[-1,+1]$. So, shift to the interval of interest:
                  $$ {a+ (b-a) sin n}_{nin mathbb{N}} .$$






                  share|cite|improve this answer









                  $endgroup$



                  Its famous that {$sin (n)}_{n=1}^infty$ is dense in $[-1,+1]$. So, shift to the interval of interest:
                  $$ {a+ (b-a) sin n}_{nin mathbb{N}} .$$







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Dec 16 '18 at 15:59









                  Behnam EsmayliBehnam Esmayli

                  1,986515




                  1,986515






























                      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%2f3042746%2fconstruct-a-sequence-subsequential-limits-of-which-form-a-range-a-b-subset%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

                      Bundesstraße 106

                      Image
                      Dieser Artikel beschreibt die Bundesstraße 106 in Deutschland. Zur gleichnamigen Straße in Österreich siehe Mölltal Straße. Vorlage:Infobox hochrangige Straße/Wartung/DE-B Bundesstraße 106 in Deutschland Karte Basisdaten Betreiber: Deutschland  Bundesrepublik Deutschland Straßenbeginn: Wismar ( 53° 54′  N , 11° 24′  O 53.900536 11.396875 ) Straßenende: Schwerin ( 53° 36′  N , 11° 24′  O 53.592785 11.404892 ) Gesamtlänge: 40,7 km Bundesland: Mecklenburg-Vorpommern Ausbauzustand: zweistreifig [1] Bundesstraße 106 in Schwerin (Umgehungsstraße) Straßenverlauf Land Mecklenburg-Vorpommern Landkreis Nordwestmecklenburg Ortsumgehung Wismar Wallensteingraben über   Dorf Mecklenburg Wallensteingraben Groß Stieten Bad Kleinen OT Niendorf Bahnstrecke Lübeck–Bad Kleinen Zickhusen Lübstorf Klein Trebbow OT Kirch Stück ...
                      Read more

                      Verónica Boquete

                      Image
                      Verónica Boquete im Trikot der Nationalmannschaft (2013) Personalia Name Verónica Boquete Giadans Geburtstag 9. April 1987 Geburtsort Santiago de Compostela, Spanien Größe 159 cm Position Mittelfeld / Sturm Juniorinnen Jahre Station 2001–2005 Xuventú Aguiño Frauen Jahre Station Spiele (Tore) 1 2005–2008 Prainsa Zaragoza 83 (40) 2008–2010 Espanyol Barcelona 53 (30) 2010 Buffalo Flash 9 0 (9) 2010 Chicago Red Stars 3 0 (1) 2010–2011 Espanyol Barcelona 26 (41) 2011 Philadelphia Independence 13 0 (5) 2011 Energija Woronesch 5 0 (1) 2012–2014 Tyresö FF 39 (14) 2014 Portland Thorns FC 15 0 (4) 2014–2015 1. FFC Frankfurt 23 0 (7) 2015–2016 FC Bayern München 9 0 (0) 2016–2018 Paris Saint-Germain 31 0 (8) 2018 Peking BG Phoenix 2019– Utah Royals FC 0 0 (0) Nationalmannschaft Jahre Auswahl ...
                      Read more

                      Ida-Boy-Ed-Garten

                      Image
                      Die Lage des Ida-Boy-Ed-Gartens, rot markiert Der Ida-Boy-Ed-Garten Der Ida-Boy-Ed-Garten, Blick von der Wakenitzmauer durch das Tor in der Stadtmauer Der Ida-Boy-Ed-Garten ist eine Straße der Lübecker Altstadt. Inhaltsverzeichnis 1 Lage 2 Geschichte 3 Literatur 4 Weblinks Lage | Der etwa 90 Meter lange Ida-Boy-Ed-Garten befindet sich an der nördlichen Spitze der Altstadtinsel, vor dem Burgtor. Die in einem Bogen durch eine Grünanlage hangaufwärts führende Straße verbindet die Wakenitzmauer mit der Burgtorbrücke. Geschichte | Der heutige Ida-Boy-Ed-Garten wurde ursprünglich als Verlängerung der Wakenitzmauer angelegt. 1903 wurde in den erhaltenen Teil der mittelalterlichen Stadtmauer, an dem die Wakenitzmauer bei der Einmündung der Kaiserstraße begann, ein Tor mit einem großen Bogen für die Durchführung der Straße und zwei kleineren seitlichen Öffnungen für Fußgänger gebrochen, um für die neu errichteten Wohnhäuser in diesem Bereich ...
                      Read more