Can a bounded number sequence be strictly ascending?Bounded Sequence ProofEvery bounded monotone sequence convergesCauchy sequences are bounded?!Why is sequence $(1+frac1n)^n+1$ descending?Show that for a sequence $p_n$ of real numbers, $limsup p_n < +∞$ iff $p_n$ is bounded above.If $a_n$ and $b_n$ are equivalent sequences and $a_n$ is bounded then so is $b_n$.Show that $s_n$ is bounded if $s_n = b_1r + b_2r^2 + … + b_nr^n$ and $0 < r < 1$.Implicit Fact about Bounded Monotone Sequence convergesBolzano-Weierstrass boundedProve that if a sequence is eventually bounded then it is bounded

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Can a bounded number sequence be strictly ascending?


Bounded Sequence ProofEvery bounded monotone sequence convergesCauchy sequences are bounded?!Why is sequence $(1+frac1n)^n+1$ descending?Show that for a sequence $p_n$ of real numbers, $limsup p_n < +∞$ iff $p_n$ is bounded above.If $a_n$ and $b_n$ are equivalent sequences and $a_n$ is bounded then so is $b_n$.Show that $s_n$ is bounded if $s_n = b_1r + b_2r^2 + … + b_nr^n$ and $0 < r < 1$.Implicit Fact about Bounded Monotone Sequence convergesBolzano-Weierstrass boundedProve that if a sequence is eventually bounded then it is bounded













1












$begingroup$


The title says it. Can a bounded number sequence be strictly ascending / descending?



I have a problem that tells me the sequence of fractional parts $(nx)_ngeq 1$ (where $x$ is given) is ascending. But I know that the sequence is bounded $[0,1)$. Thus, shouldn’t the sequence stop ascending at a point? Please show me a proof or something.






sequences-and-series







share|cite|improve this question











$endgroup$











  • $begingroup$
    A series deals with summation. A sequence deals with individual elements.
    $endgroup$
    – Subhasis Biswas
    2 hours ago















1












$begingroup$


The title says it. Can a bounded number sequence be strictly ascending / descending?



I have a problem that tells me the sequence of fractional parts $(nx)_ngeq 1$ (where $x$ is given) is ascending. But I know that the sequence is bounded $[0,1)$. Thus, shouldn’t the sequence stop ascending at a point? Please show me a proof or something.






sequences-and-series







share|cite|improve this question











$endgroup$











  • $begingroup$
    A series deals with summation. A sequence deals with individual elements.
    $endgroup$
    – Subhasis Biswas
    2 hours ago













1












1








1





$begingroup$


The title says it. Can a bounded number sequence be strictly ascending / descending?



I have a problem that tells me the sequence of fractional parts $(nx)_ngeq 1$ (where $x$ is given) is ascending. But I know that the sequence is bounded $[0,1)$. Thus, shouldn’t the sequence stop ascending at a point? Please show me a proof or something.






sequences-and-series







share|cite|improve this question











$endgroup$




The title says it. Can a bounded number sequence be strictly ascending / descending?



I have a problem that tells me the sequence of fractional parts $(nx)_ngeq 1$ (where $x$ is given) is ascending. But I know that the sequence is bounded $[0,1)$. Thus, shouldn’t the sequence stop ascending at a point? Please show me a proof or something.






sequences-and-series




sequences-and-series






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 2 hours ago







furfur

















asked 2 hours ago









furfurfurfur

619




619











  • $begingroup$
    A series deals with summation. A sequence deals with individual elements.
    $endgroup$
    – Subhasis Biswas
    2 hours ago
















  • $begingroup$
    A series deals with summation. A sequence deals with individual elements.
    $endgroup$
    – Subhasis Biswas
    2 hours ago















$begingroup$
A series deals with summation. A sequence deals with individual elements.
$endgroup$
– Subhasis Biswas
2 hours ago




$begingroup$
A series deals with summation. A sequence deals with individual elements.
$endgroup$
– Subhasis Biswas
2 hours ago










2 Answers
2






active

oldest

votes


















6












$begingroup$

I presume you mean sequence, not series. For example, the sequence $1 - 1/n$ is bounded and strictly increasing.






share|cite|improve this answer









$endgroup$




















    6












    $begingroup$

    Sure it can.



    Hint



    $0.9 ;,; 0.99 ;,; 0.999 ;,; ldots$






    share|cite|improve this answer









    $endgroup$












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      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      6












      $begingroup$

      I presume you mean sequence, not series. For example, the sequence $1 - 1/n$ is bounded and strictly increasing.






      share|cite|improve this answer









      $endgroup$

















        6












        $begingroup$

        I presume you mean sequence, not series. For example, the sequence $1 - 1/n$ is bounded and strictly increasing.






        share|cite|improve this answer









        $endgroup$















          6












          6








          6





          $begingroup$

          I presume you mean sequence, not series. For example, the sequence $1 - 1/n$ is bounded and strictly increasing.






          share|cite|improve this answer









          $endgroup$



          I presume you mean sequence, not series. For example, the sequence $1 - 1/n$ is bounded and strictly increasing.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 2 hours ago









          Robert IsraelRobert Israel

          327k23216469




          327k23216469





















              6












              $begingroup$

              Sure it can.



              Hint



              $0.9 ;,; 0.99 ;,; 0.999 ;,; ldots$






              share|cite|improve this answer









              $endgroup$

















                6












                $begingroup$

                Sure it can.



                Hint



                $0.9 ;,; 0.99 ;,; 0.999 ;,; ldots$






                share|cite|improve this answer









                $endgroup$















                  6












                  6








                  6





                  $begingroup$

                  Sure it can.



                  Hint



                  $0.9 ;,; 0.99 ;,; 0.999 ;,; ldots$






                  share|cite|improve this answer









                  $endgroup$



                  Sure it can.



                  Hint



                  $0.9 ;,; 0.99 ;,; 0.999 ;,; ldots$







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 2 hours ago









                  StackTDStackTD

                  23.4k2153




                  23.4k2153



























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