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









    $endgroup$













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      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






























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