$(x_n)_{n in mathbb{N}}$ is cauchy if and only if $d(x_{n_k},x_{m_k}) rightarrow 0$ for any two sequences...
I want to know if the following assertion is correct:
If $(M,d)$ is a metric space, then $(x_n)_{n in mathbb{N}}$ is cauchy if and only if $d(x_{n_k},x_{m_k}) rightarrow 0$ for any two sequences of natural numbers $n_k, m_k rightarrow infty$
Of course the direction from left to right is true. But I am unsure of the direction from right to left. Can anyone give a proof or state a counterexample?
Thanks a lot in advance!
real-analysis sequences-and-series analysis
asked Oct 23 at 22:52
vaoy
51729
add a comment |
I want to know if the following assertion is correct:
If $(M,d)$ is a metric space, then $(x_n)_{n in mathbb{N}}$ is cauchy if and only if $d(x_{n_k},x_{m_k}) rightarrow 0$ for any two sequences of natural numbers $n_k, m_k rightarrow infty$
Of course the direction from left to right is true. But I am unsure of the direction from right to left. Can anyone give a proof or state a counterexample?
Thanks a lot in advance!
real-analysis sequences-and-series analysis
asked Oct 23 at 22:52
vaoy
51729
1
I'd assume the sequence of the distances to tend to $0$ instead of $infty$ for a Cauchy series!
– Ingix
Oct 23 at 23:00
Oh yes, that was a typo. Do you know how to check if this is true?
– vaoy
Oct 23 at 23:05
Just the negation of $(x_n)$ being Cauchy. You will get the implication from right to left easily.
– Kavi Rama Murthy
Oct 23 at 23:50
add a comment |
I want to know if the following assertion is correct:
If $(M,d)$ is a metric space, then $(x_n)_{n in mathbb{N}}$ is cauchy if and only if $d(x_{n_k},x_{m_k}) rightarrow 0$ for any two sequences of natural numbers $n_k, m_k rightarrow infty$
Of course the direction from left to right is true. But I am unsure of the direction from right to left. Can anyone give a proof or state a counterexample?
Thanks a lot in advance!
real-analysis sequences-and-series analysis
asked Oct 23 at 22:52
vaoy
51729
I want to know if the following assertion is correct:
If $(M,d)$ is a metric space, then $(x_n)_{n in mathbb{N}}$ is cauchy if and only if $d(x_{n_k},x_{m_k}) rightarrow 0$ for any two sequences of natural numbers $n_k, m_k rightarrow infty$
Of course the direction from left to right is true. But I am unsure of the direction from right to left. Can anyone give a proof or state a counterexample?
Thanks a lot in advance!
real-analysis sequences-and-series analysis
real-analysis sequences-and-series analysis
asked Oct 23 at 22:52
vaoy
51729
asked Oct 23 at 22:52
vaoy
51729
edited Oct 23 at 23:03
asked Oct 23 at 22:52
vaoy
51729
asked Oct 23 at 22:52
vaoy
51729
asked Oct 23 at 22:52
vaoy
51729
51729
1
I'd assume the sequence of the distances to tend to $0$ instead of $infty$ for a Cauchy series!
– Ingix
Oct 23 at 23:00
Oh yes, that was a typo. Do you know how to check if this is true?
– vaoy
Oct 23 at 23:05
Just the negation of $(x_n)$ being Cauchy. You will get the implication from right to left easily.
– Kavi Rama Murthy
Oct 23 at 23:50
add a comment |
1
I'd assume the sequence of the distances to tend to $0$ instead of $infty$ for a Cauchy series!
– Ingix
Oct 23 at 23:00
Oh yes, that was a typo. Do you know how to check if this is true?
– vaoy
Oct 23 at 23:05
Just the negation of $(x_n)$ being Cauchy. You will get the implication from right to left easily.
– Kavi Rama Murthy
Oct 23 at 23:50
1
1
I'd assume the sequence of the distances to tend to $0$ instead of $infty$ for a Cauchy series!
– Ingix
Oct 23 at 23:00
I'd assume the sequence of the distances to tend to $0$ instead of $infty$ for a Cauchy series!
– Ingix
Oct 23 at 23:00
Oh yes, that was a typo. Do you know how to check if this is true?
– vaoy
Oct 23 at 23:05
Oh yes, that was a typo. Do you know how to check if this is true?
– vaoy
Oct 23 at 23:05
Just the negation of $(x_n)$ being Cauchy. You will get the implication from right to left easily.
– Kavi Rama Murthy
Oct 23 at 23:50
Just the negation of $(x_n)$ being Cauchy. You will get the implication from right to left easily.
– Kavi Rama Murthy
Oct 23 at 23:50
add a comment |
1 Answer
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Since the right side is true for any sequences $n_k$ and $m_k$, simply take $n_k=k$ and $m_k=k+1$ and conclude.
answered Nov 30 at 18:47
Mostafa Ayaz
13.7k3836
add a comment |
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Since the right side is true for any sequences $n_k$ and $m_k$, simply take $n_k=k$ and $m_k=k+1$ and conclude.
answered Nov 30 at 18:47
Mostafa Ayaz
13.7k3836
add a comment |
Since the right side is true for any sequences $n_k$ and $m_k$, simply take $n_k=k$ and $m_k=k+1$ and conclude.
answered Nov 30 at 18:47
Mostafa Ayaz
13.7k3836
add a comment |
Since the right side is true for any sequences $n_k$ and $m_k$, simply take $n_k=k$ and $m_k=k+1$ and conclude.
answered Nov 30 at 18:47
Mostafa Ayaz
13.7k3836
Since the right side is true for any sequences $n_k$ and $m_k$, simply take $n_k=k$ and $m_k=k+1$ and conclude.
answered Nov 30 at 18:47
Mostafa Ayaz
13.7k3836
answered Nov 30 at 18:47
Mostafa Ayaz
13.7k3836
answered Nov 30 at 18:47
Mostafa Ayaz
13.7k3836
answered Nov 30 at 18:47
Mostafa Ayaz
13.7k3836
13.7k3836
add a comment |
add a comment |
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1
I'd assume the sequence of the distances to tend to $0$ instead of $infty$ for a Cauchy series!
– Ingix
Oct 23 at 23:00
Oh yes, that was a typo. Do you know how to check if this is true?
– vaoy
Oct 23 at 23:05
Just the negation of $(x_n)$ being Cauchy. You will get the implication from right to left easily.
– Kavi Rama Murthy
Oct 23 at 23:50