Prove sequence in a complete metric space converges if the series of distances converges. The 2019 Stack Overflow Developer Survey Results Are InFor a sequence of non negative numbers, if the series converges, then the series of the sequence raised to p also converges if p>=1If two sequences converge in a metric space, the sequence of the distances converges to the distance of limits of the sequences.Prove that the space of sequences under this metric is complete and compact.Convergence in a complete metric spaceCauchy sequence in compact metric space converges; incorrect proof?Is a bounded closed subset on a locally compact metric space complete?How to show the Jungle River Metric is CompleteShow that the space $(X,d)$ is not complete and prove its completionProve/Disprove: If $M$ is a complete metric space then the limit points are in $M$The identity map for the euclidean norm (i.e. absolute value) is a complete metric space explanation
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Prove sequence in a complete metric space converges if the series of distances converges.
The 2019 Stack Overflow Developer Survey Results Are InFor a sequence of non negative numbers, if the series converges, then the series of the sequence raised to p also converges if p>=1If two sequences converge in a metric space, the sequence of the distances converges to the distance of limits of the sequences.Prove that the space of sequences under this metric is complete and compact.Convergence in a complete metric spaceCauchy sequence in compact metric space converges; incorrect proof?Is a bounded closed subset on a locally compact metric space complete?How to show the Jungle River Metric is CompleteShow that the space $(X,d)$ is not complete and prove its completionProve/Disprove: If $M$ is a complete metric space then the limit points are in $M$The identity map for the euclidean norm (i.e. absolute value) is a complete metric space explanation
$begingroup$
Assume that X is complete, and let $(p_n)$ be a sequence in X. Assume that $sumlimits_n=1^infty d(p_n, p_n+1)$ converges. Prove that $(p_n)$ converges.
(X,d) is a metric space.
Don't quite know how to start. Any help/hints appreciated!
real-analysis sequences-and-series convergence complete-spaces
edited Mar 30 at 6:15
Martin Sleziak
45k10122277
asked Nov 25 '13 at 4:18
akeenlogicianakeenlogician
172214
$endgroup$
add a comment |
$begingroup$
Assume that X is complete, and let $(p_n)$ be a sequence in X. Assume that $sumlimits_n=1^infty d(p_n, p_n+1)$ converges. Prove that $(p_n)$ converges.
(X,d) is a metric space.
Don't quite know how to start. Any help/hints appreciated!
real-analysis sequences-and-series convergence complete-spaces
edited Mar 30 at 6:15
Martin Sleziak
45k10122277
asked Nov 25 '13 at 4:18
akeenlogicianakeenlogician
172214
$endgroup$
add a comment |
$begingroup$
Assume that X is complete, and let $(p_n)$ be a sequence in X. Assume that $sumlimits_n=1^infty d(p_n, p_n+1)$ converges. Prove that $(p_n)$ converges.
(X,d) is a metric space.
Don't quite know how to start. Any help/hints appreciated!
real-analysis sequences-and-series convergence complete-spaces
edited Mar 30 at 6:15
Martin Sleziak
45k10122277
asked Nov 25 '13 at 4:18
akeenlogicianakeenlogician
172214
$endgroup$
Assume that X is complete, and let $(p_n)$ be a sequence in X. Assume that $sumlimits_n=1^infty d(p_n, p_n+1)$ converges. Prove that $(p_n)$ converges.
(X,d) is a metric space.
Don't quite know how to start. Any help/hints appreciated!
real-analysis sequences-and-series convergence complete-spaces
real-analysis sequences-and-series convergence complete-spaces
edited Mar 30 at 6:15
Martin Sleziak
45k10122277
asked Nov 25 '13 at 4:18
akeenlogicianakeenlogician
172214
edited Mar 30 at 6:15
Martin Sleziak
45k10122277
asked Nov 25 '13 at 4:18
akeenlogicianakeenlogician
172214
edited Mar 30 at 6:15
Martin Sleziak
45k10122277
edited Mar 30 at 6:15
Martin Sleziak
45k10122277
edited Mar 30 at 6:15
Martin Sleziak
45k10122277
45k10122277
asked Nov 25 '13 at 4:18
akeenlogicianakeenlogician
172214
asked Nov 25 '13 at 4:18
akeenlogicianakeenlogician
172214
asked Nov 25 '13 at 4:18
akeenlogicianakeenlogician
172214
172214
add a comment |
add a comment |
2 Answers
2
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oldest
votes
$begingroup$
How to start: look at the definition of "complete metric space". Every Cauchy sequence converges. OK, we have to show that $(p_n)$ is a Cauchy sequence. Using, somehow, the fact that the series of distances converges.
Next, consider the definition of a Cauchy sequence. Given a positive $varepsilon$ we have to show that, if $n$ is big enough, then $d(p_n,p_m)ltvarepsilon$ for all $mgt n$. Right? Is that how the Cauchy criterion is stated in your book?
Hmm. How do we get from $d(p_n,p_n+1)$ to $d(p_n,p_m)$?? The triangle inequality!! If $m=n+k$ then$$d(p_n,p_m)=d(p_n,p_n+k)le d(p_n,p_n+1)+d(p_n+1,p_n+2)+cdots+d(p_n+k-1,p_n+k)le d(p_n,p_n+1)+d(p_n+1,p_n+2)+d(p_n+2,p_n+3)+cdots$$
Hmm. The remainder of a convergent series. Goes to zero, doesn't it?
answered Nov 25 '13 at 4:40
bofbof
52.6k559121
$endgroup$
add a comment |
$begingroup$
HINT: Show that if $sum_nge 1d(p_n,p_n+1)$ converges, then $langle p_n:ninBbb Z^+rangle$ is a Cauchy sequence. If that’s not quite enough, there’s a further hint in the spoiler-protected block below.
Further HINT: Show that the hypothesis implies that $sum_nge md(p_n,p_n+1)to 0$ as $mtoinfty$.
answered Nov 25 '13 at 4:29
Brian M. ScottBrian M. Scott
460k40518919
$endgroup$
add a comment |
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2 Answers
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2 Answers
2
active
oldest
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active
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votes
$begingroup$
How to start: look at the definition of "complete metric space". Every Cauchy sequence converges. OK, we have to show that $(p_n)$ is a Cauchy sequence. Using, somehow, the fact that the series of distances converges.
Next, consider the definition of a Cauchy sequence. Given a positive $varepsilon$ we have to show that, if $n$ is big enough, then $d(p_n,p_m)ltvarepsilon$ for all $mgt n$. Right? Is that how the Cauchy criterion is stated in your book?
Hmm. How do we get from $d(p_n,p_n+1)$ to $d(p_n,p_m)$?? The triangle inequality!! If $m=n+k$ then$$d(p_n,p_m)=d(p_n,p_n+k)le d(p_n,p_n+1)+d(p_n+1,p_n+2)+cdots+d(p_n+k-1,p_n+k)le d(p_n,p_n+1)+d(p_n+1,p_n+2)+d(p_n+2,p_n+3)+cdots$$
Hmm. The remainder of a convergent series. Goes to zero, doesn't it?
answered Nov 25 '13 at 4:40
bofbof
52.6k559121
$endgroup$
add a comment |
$begingroup$
How to start: look at the definition of "complete metric space". Every Cauchy sequence converges. OK, we have to show that $(p_n)$ is a Cauchy sequence. Using, somehow, the fact that the series of distances converges.
Next, consider the definition of a Cauchy sequence. Given a positive $varepsilon$ we have to show that, if $n$ is big enough, then $d(p_n,p_m)ltvarepsilon$ for all $mgt n$. Right? Is that how the Cauchy criterion is stated in your book?
Hmm. How do we get from $d(p_n,p_n+1)$ to $d(p_n,p_m)$?? The triangle inequality!! If $m=n+k$ then$$d(p_n,p_m)=d(p_n,p_n+k)le d(p_n,p_n+1)+d(p_n+1,p_n+2)+cdots+d(p_n+k-1,p_n+k)le d(p_n,p_n+1)+d(p_n+1,p_n+2)+d(p_n+2,p_n+3)+cdots$$
Hmm. The remainder of a convergent series. Goes to zero, doesn't it?
answered Nov 25 '13 at 4:40
bofbof
52.6k559121
$endgroup$
add a comment |
$begingroup$
How to start: look at the definition of "complete metric space". Every Cauchy sequence converges. OK, we have to show that $(p_n)$ is a Cauchy sequence. Using, somehow, the fact that the series of distances converges.
Next, consider the definition of a Cauchy sequence. Given a positive $varepsilon$ we have to show that, if $n$ is big enough, then $d(p_n,p_m)ltvarepsilon$ for all $mgt n$. Right? Is that how the Cauchy criterion is stated in your book?
Hmm. How do we get from $d(p_n,p_n+1)$ to $d(p_n,p_m)$?? The triangle inequality!! If $m=n+k$ then$$d(p_n,p_m)=d(p_n,p_n+k)le d(p_n,p_n+1)+d(p_n+1,p_n+2)+cdots+d(p_n+k-1,p_n+k)le d(p_n,p_n+1)+d(p_n+1,p_n+2)+d(p_n+2,p_n+3)+cdots$$
Hmm. The remainder of a convergent series. Goes to zero, doesn't it?
answered Nov 25 '13 at 4:40
bofbof
52.6k559121
$endgroup$
How to start: look at the definition of "complete metric space". Every Cauchy sequence converges. OK, we have to show that $(p_n)$ is a Cauchy sequence. Using, somehow, the fact that the series of distances converges.
Next, consider the definition of a Cauchy sequence. Given a positive $varepsilon$ we have to show that, if $n$ is big enough, then $d(p_n,p_m)ltvarepsilon$ for all $mgt n$. Right? Is that how the Cauchy criterion is stated in your book?
Hmm. How do we get from $d(p_n,p_n+1)$ to $d(p_n,p_m)$?? The triangle inequality!! If $m=n+k$ then$$d(p_n,p_m)=d(p_n,p_n+k)le d(p_n,p_n+1)+d(p_n+1,p_n+2)+cdots+d(p_n+k-1,p_n+k)le d(p_n,p_n+1)+d(p_n+1,p_n+2)+d(p_n+2,p_n+3)+cdots$$
Hmm. The remainder of a convergent series. Goes to zero, doesn't it?
answered Nov 25 '13 at 4:40
bofbof
52.6k559121
answered Nov 25 '13 at 4:40
bofbof
52.6k559121
answered Nov 25 '13 at 4:40
bofbof
52.6k559121
answered Nov 25 '13 at 4:40
bofbof
52.6k559121
52.6k559121
add a comment |
add a comment |
$begingroup$
HINT: Show that if $sum_nge 1d(p_n,p_n+1)$ converges, then $langle p_n:ninBbb Z^+rangle$ is a Cauchy sequence. If that’s not quite enough, there’s a further hint in the spoiler-protected block below.
Further HINT: Show that the hypothesis implies that $sum_nge md(p_n,p_n+1)to 0$ as $mtoinfty$.
answered Nov 25 '13 at 4:29
Brian M. ScottBrian M. Scott
460k40518919
$endgroup$
add a comment |
$begingroup$
HINT: Show that if $sum_nge 1d(p_n,p_n+1)$ converges, then $langle p_n:ninBbb Z^+rangle$ is a Cauchy sequence. If that’s not quite enough, there’s a further hint in the spoiler-protected block below.
Further HINT: Show that the hypothesis implies that $sum_nge md(p_n,p_n+1)to 0$ as $mtoinfty$.
answered Nov 25 '13 at 4:29
Brian M. ScottBrian M. Scott
460k40518919
$endgroup$
add a comment |
$begingroup$
HINT: Show that if $sum_nge 1d(p_n,p_n+1)$ converges, then $langle p_n:ninBbb Z^+rangle$ is a Cauchy sequence. If that’s not quite enough, there’s a further hint in the spoiler-protected block below.
Further HINT: Show that the hypothesis implies that $sum_nge md(p_n,p_n+1)to 0$ as $mtoinfty$.
answered Nov 25 '13 at 4:29
Brian M. ScottBrian M. Scott
460k40518919
$endgroup$
HINT: Show that if $sum_nge 1d(p_n,p_n+1)$ converges, then $langle p_n:ninBbb Z^+rangle$ is a Cauchy sequence. If that’s not quite enough, there’s a further hint in the spoiler-protected block below.
Further HINT: Show that the hypothesis implies that $sum_nge md(p_n,p_n+1)to 0$ as $mtoinfty$.
answered Nov 25 '13 at 4:29
Brian M. ScottBrian M. Scott
460k40518919
answered Nov 25 '13 at 4:29
Brian M. ScottBrian M. Scott
460k40518919
answered Nov 25 '13 at 4:29
Brian M. ScottBrian M. Scott
460k40518919
answered Nov 25 '13 at 4:29
Brian M. ScottBrian M. Scott
460k40518919
460k40518919
add a comment |
add a comment |
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