Sequence in the complete metric space [closed]











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$f_n: [0,1]to mathbb{R}$



$f_n(x)=x^n$ in the complete metric space of bounded functions equipped with the supremum metric.



Is it a Cauchy sequence?










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closed as off-topic by Scientifica, user10354138, Chinnapparaj R, Lee David Chung Lin, Micah Nov 18 at 4:36


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  • What is $;limlimits_{ntoinfty} x^n;$ ...: (1) when $;0le x<1;$ ? When $;x=1;$ ?
    – DonAntonio
    Nov 17 at 17:16

















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0
down vote

favorite












$f_n: [0,1]to mathbb{R}$



$f_n(x)=x^n$ in the complete metric space of bounded functions equipped with the supremum metric.



Is it a Cauchy sequence?










share|cite|improve this question















closed as off-topic by Scientifica, user10354138, Chinnapparaj R, Lee David Chung Lin, Micah Nov 18 at 4:36


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Scientifica, user10354138, Chinnapparaj R, Lee David Chung Lin, Micah

If this question can be reworded to fit the rules in the help center, please edit the question.













  • What is $;limlimits_{ntoinfty} x^n;$ ...: (1) when $;0le x<1;$ ? When $;x=1;$ ?
    – DonAntonio
    Nov 17 at 17:16















up vote
0
down vote

favorite









up vote
0
down vote

favorite











$f_n: [0,1]to mathbb{R}$



$f_n(x)=x^n$ in the complete metric space of bounded functions equipped with the supremum metric.



Is it a Cauchy sequence?










share|cite|improve this question















$f_n: [0,1]to mathbb{R}$



$f_n(x)=x^n$ in the complete metric space of bounded functions equipped with the supremum metric.



Is it a Cauchy sequence?







real-analysis analysis metric-spaces






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edited Nov 17 at 17:42









Henno Brandsma

102k345108




102k345108










asked Nov 17 at 17:12









Maria Julia

14




14




closed as off-topic by Scientifica, user10354138, Chinnapparaj R, Lee David Chung Lin, Micah Nov 18 at 4:36


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Scientifica, user10354138, Chinnapparaj R, Lee David Chung Lin, Micah

If this question can be reworded to fit the rules in the help center, please edit the question.




closed as off-topic by Scientifica, user10354138, Chinnapparaj R, Lee David Chung Lin, Micah Nov 18 at 4:36


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Scientifica, user10354138, Chinnapparaj R, Lee David Chung Lin, Micah

If this question can be reworded to fit the rules in the help center, please edit the question.












  • What is $;limlimits_{ntoinfty} x^n;$ ...: (1) when $;0le x<1;$ ? When $;x=1;$ ?
    – DonAntonio
    Nov 17 at 17:16




















  • What is $;limlimits_{ntoinfty} x^n;$ ...: (1) when $;0le x<1;$ ? When $;x=1;$ ?
    – DonAntonio
    Nov 17 at 17:16


















What is $;limlimits_{ntoinfty} x^n;$ ...: (1) when $;0le x<1;$ ? When $;x=1;$ ?
– DonAntonio
Nov 17 at 17:16






What is $;limlimits_{ntoinfty} x^n;$ ...: (1) when $;0le x<1;$ ? When $;x=1;$ ?
– DonAntonio
Nov 17 at 17:16












1 Answer
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It can't be Cauchy. The normed space you considered is complete, hence $f_n$ would have an uniform limit if it was Cauchy. On the other hand the pointwise limit of $f_n$ is not continuous and so it can't be uniform limit of continuous functions.






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    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    1
    down vote













    It can't be Cauchy. The normed space you considered is complete, hence $f_n$ would have an uniform limit if it was Cauchy. On the other hand the pointwise limit of $f_n$ is not continuous and so it can't be uniform limit of continuous functions.






    share|cite|improve this answer

























      up vote
      1
      down vote













      It can't be Cauchy. The normed space you considered is complete, hence $f_n$ would have an uniform limit if it was Cauchy. On the other hand the pointwise limit of $f_n$ is not continuous and so it can't be uniform limit of continuous functions.






      share|cite|improve this answer























        up vote
        1
        down vote










        up vote
        1
        down vote









        It can't be Cauchy. The normed space you considered is complete, hence $f_n$ would have an uniform limit if it was Cauchy. On the other hand the pointwise limit of $f_n$ is not continuous and so it can't be uniform limit of continuous functions.






        share|cite|improve this answer












        It can't be Cauchy. The normed space you considered is complete, hence $f_n$ would have an uniform limit if it was Cauchy. On the other hand the pointwise limit of $f_n$ is not continuous and so it can't be uniform limit of continuous functions.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 17 at 20:51









        Marco

        1909




        1909















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