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?
real-analysis analysis metric-spaces
edited Nov 17 at 17:42
Henno Brandsma
102k345108
asked Nov 17 at 17:12
Maria Julia
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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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$f_n: [0,1]to mathbb{R}$
$f_n(x)=x^n$ in the complete metric space of bounded functions equipped with the supremum metric.
Is it a Cauchy sequence?
real-analysis analysis metric-spaces
edited Nov 17 at 17:42
Henno Brandsma
102k345108
asked Nov 17 at 17:12
Maria Julia
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.
What is $;limlimits_{ntoinfty} x^n;$ ...: (1) when $;0le x<1;$ ? When $;x=1;$ ?
– DonAntonio
Nov 17 at 17:16
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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?
real-analysis analysis metric-spaces
edited Nov 17 at 17:42
Henno Brandsma
102k345108
asked Nov 17 at 17:12
Maria Julia
14
$f_n: [0,1]to mathbb{R}$
$f_n(x)=x^n$ in the complete metric space of bounded functions equipped with the supremum metric.
Is it a Cauchy sequence?
real-analysis analysis metric-spaces
real-analysis analysis metric-spaces
edited Nov 17 at 17:42
Henno Brandsma
102k345108
asked Nov 17 at 17:12
Maria Julia
14
edited Nov 17 at 17:42
Henno Brandsma
102k345108
asked Nov 17 at 17:12
Maria Julia
14
edited Nov 17 at 17:42
Henno Brandsma
102k345108
edited Nov 17 at 17:42
Henno Brandsma
102k345108
edited Nov 17 at 17:42
Henno Brandsma
102k345108
102k345108
asked Nov 17 at 17:12
Maria Julia
14
asked Nov 17 at 17:12
Maria Julia
14
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
add a comment |
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
add a comment |
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.
answered Nov 17 at 20:51
Marco
1909
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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.
answered Nov 17 at 20:51
Marco
1909
add a comment |
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.
answered Nov 17 at 20:51
Marco
1909
add a comment |
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.
answered Nov 17 at 20:51
Marco
1909
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.
answered Nov 17 at 20:51
Marco
1909
answered Nov 17 at 20:51
Marco
1909
answered Nov 17 at 20:51
Marco
1909
answered Nov 17 at 20:51
Marco
1909
1909
add a comment |
add a comment |
What is $;limlimits_{ntoinfty} x^n;$ ...: (1) when $;0le x<1;$ ? When $;x=1;$ ?
– DonAntonio
Nov 17 at 17:16