limit of sequence involving the fractional part

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Using the pigeonhole principle, any sequence of the form $(fracnr)_ngeq1$ where $r$ is an irrational number is dense in the unit interval. Then prove that the following limit does not exit in $[0;infty]$



$$lim_ntoinftynbiggfracnrbigg$$




sequences-and-series limits fractional-part




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  • Your thoughts on this problem, how far your got, would be appreciated.
    – mvw
    Jul 25 at 21:10






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    As the sequence $(fracnr)_ngeq1$ is dense in the unit interval so we can get infinite sub-sequences of $(nfracnr)_ngeq1$ that blow to $infty$ yet the question is what gonna happen if one chooses a subsequence of $(fracnr)_ngeq1$ that converges to $0$, in such a case the speed of convergence of subsequence of $(fracnr)_ngeq1$ will affect the limit associated of subsequence of $(nfracnr)_ngeq1$.
    – Kays Tomy
    Jul 25 at 21:18










  • Are you familiar with continued fractions?
    – Daniel Fischer♦
    Jul 25 at 22:07














up vote
2
down vote

favorite
1












Using the pigeonhole principle, any sequence of the form $(fracnr)_ngeq1$ where $r$ is an irrational number is dense in the unit interval. Then prove that the following limit does not exit in $[0;infty]$



$$lim_ntoinftynbiggfracnrbigg$$




sequences-and-series limits fractional-part




share|cite|improve this question



















  • Your thoughts on this problem, how far your got, would be appreciated.
    – mvw
    Jul 25 at 21:10






  • 1




    As the sequence $(fracnr)_ngeq1$ is dense in the unit interval so we can get infinite sub-sequences of $(nfracnr)_ngeq1$ that blow to $infty$ yet the question is what gonna happen if one chooses a subsequence of $(fracnr)_ngeq1$ that converges to $0$, in such a case the speed of convergence of subsequence of $(fracnr)_ngeq1$ will affect the limit associated of subsequence of $(nfracnr)_ngeq1$.
    – Kays Tomy
    Jul 25 at 21:18










  • Are you familiar with continued fractions?
    – Daniel Fischer♦
    Jul 25 at 22:07












up vote
2
down vote

favorite
1









up vote
2
down vote

favorite
1






1





Using the pigeonhole principle, any sequence of the form $(fracnr)_ngeq1$ where $r$ is an irrational number is dense in the unit interval. Then prove that the following limit does not exit in $[0;infty]$



$$lim_ntoinftynbiggfracnrbigg$$




sequences-and-series limits fractional-part




share|cite|improve this question











Using the pigeonhole principle, any sequence of the form $(fracnr)_ngeq1$ where $r$ is an irrational number is dense in the unit interval. Then prove that the following limit does not exit in $[0;infty]$



$$lim_ntoinftynbiggfracnrbigg$$





sequences-and-series limits fractional-part





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 25 at 20:34









Kays Tomy

954




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  • Your thoughts on this problem, how far your got, would be appreciated.
    – mvw
    Jul 25 at 21:10






  • 1




    As the sequence $(fracnr)_ngeq1$ is dense in the unit interval so we can get infinite sub-sequences of $(nfracnr)_ngeq1$ that blow to $infty$ yet the question is what gonna happen if one chooses a subsequence of $(fracnr)_ngeq1$ that converges to $0$, in such a case the speed of convergence of subsequence of $(fracnr)_ngeq1$ will affect the limit associated of subsequence of $(nfracnr)_ngeq1$.
    – Kays Tomy
    Jul 25 at 21:18










  • Are you familiar with continued fractions?
    – Daniel Fischer♦
    Jul 25 at 22:07
















  • Your thoughts on this problem, how far your got, would be appreciated.
    – mvw
    Jul 25 at 21:10






  • 1




    As the sequence $(fracnr)_ngeq1$ is dense in the unit interval so we can get infinite sub-sequences of $(nfracnr)_ngeq1$ that blow to $infty$ yet the question is what gonna happen if one chooses a subsequence of $(fracnr)_ngeq1$ that converges to $0$, in such a case the speed of convergence of subsequence of $(fracnr)_ngeq1$ will affect the limit associated of subsequence of $(nfracnr)_ngeq1$.
    – Kays Tomy
    Jul 25 at 21:18










  • Are you familiar with continued fractions?
    – Daniel Fischer♦
    Jul 25 at 22:07















Your thoughts on this problem, how far your got, would be appreciated.
– mvw
Jul 25 at 21:10




Your thoughts on this problem, how far your got, would be appreciated.
– mvw
Jul 25 at 21:10




1




1




As the sequence $(fracnr)_ngeq1$ is dense in the unit interval so we can get infinite sub-sequences of $(nfracnr)_ngeq1$ that blow to $infty$ yet the question is what gonna happen if one chooses a subsequence of $(fracnr)_ngeq1$ that converges to $0$, in such a case the speed of convergence of subsequence of $(fracnr)_ngeq1$ will affect the limit associated of subsequence of $(nfracnr)_ngeq1$.
– Kays Tomy
Jul 25 at 21:18




As the sequence $(fracnr)_ngeq1$ is dense in the unit interval so we can get infinite sub-sequences of $(nfracnr)_ngeq1$ that blow to $infty$ yet the question is what gonna happen if one chooses a subsequence of $(fracnr)_ngeq1$ that converges to $0$, in such a case the speed of convergence of subsequence of $(fracnr)_ngeq1$ will affect the limit associated of subsequence of $(nfracnr)_ngeq1$.
– Kays Tomy
Jul 25 at 21:18












Are you familiar with continued fractions?
– Daniel Fischer♦
Jul 25 at 22:07




Are you familiar with continued fractions?
– Daniel Fischer♦
Jul 25 at 22:07















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