C($mathbb{R}$) is complete under the discrete metric.
$begingroup$
Examine if the space $X=C(mathbb{R})$ endowed with the discrete metric $d_0$ is complete or not.
We know that $(X,d_0)$ is complete if every Cauchy sequence is finally constant. We want to construct a function $f(x_n),Cauchy$ such that $f(x_n)rightarrow f(x_0)$ and also $f$ is continuous.
Is this gonna work?
metric-spaces cauchy-sequences complete-spaces
edited Jan 7 at 13:49
José Carlos Santos
153k22123225
asked Oct 11 '18 at 11:29
argiriskarargiriskar
1409
$endgroup$
add a comment |
$begingroup$
Examine if the space $X=C(mathbb{R})$ endowed with the discrete metric $d_0$ is complete or not.
We know that $(X,d_0)$ is complete if every Cauchy sequence is finally constant. We want to construct a function $f(x_n),Cauchy$ such that $f(x_n)rightarrow f(x_0)$ and also $f$ is continuous.
Is this gonna work?
metric-spaces cauchy-sequences complete-spaces
edited Jan 7 at 13:49
José Carlos Santos
153k22123225
asked Oct 11 '18 at 11:29
argiriskarargiriskar
1409
$endgroup$
add a comment |
$begingroup$
Examine if the space $X=C(mathbb{R})$ endowed with the discrete metric $d_0$ is complete or not.
We know that $(X,d_0)$ is complete if every Cauchy sequence is finally constant. We want to construct a function $f(x_n),Cauchy$ such that $f(x_n)rightarrow f(x_0)$ and also $f$ is continuous.
Is this gonna work?
metric-spaces cauchy-sequences complete-spaces
edited Jan 7 at 13:49
José Carlos Santos
153k22123225
asked Oct 11 '18 at 11:29
argiriskarargiriskar
1409
$endgroup$
Examine if the space $X=C(mathbb{R})$ endowed with the discrete metric $d_0$ is complete or not.
We know that $(X,d_0)$ is complete if every Cauchy sequence is finally constant. We want to construct a function $f(x_n),Cauchy$ such that $f(x_n)rightarrow f(x_0)$ and also $f$ is continuous.
Is this gonna work?
metric-spaces cauchy-sequences complete-spaces
metric-spaces cauchy-sequences complete-spaces
edited Jan 7 at 13:49
José Carlos Santos
153k22123225
asked Oct 11 '18 at 11:29
argiriskarargiriskar
1409
edited Jan 7 at 13:49
José Carlos Santos
153k22123225
asked Oct 11 '18 at 11:29
argiriskarargiriskar
1409
edited Jan 7 at 13:49
José Carlos Santos
153k22123225
edited Jan 7 at 13:49
José Carlos Santos
153k22123225
edited Jan 7 at 13:49
José Carlos Santos
153k22123225
153k22123225
asked Oct 11 '18 at 11:29
argiriskarargiriskar
1409
asked Oct 11 '18 at 11:29
argiriskarargiriskar
1409
asked Oct 11 '18 at 11:29
argiriskarargiriskar
1409
1409
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
If you endow any set with that discrete metric, it becomes a complete metric space, because any Cauchy sequence is constant then, if $n$ is large enough.
answered Oct 11 '18 at 11:31
José Carlos SantosJosé Carlos Santos
153k22123225
$endgroup$
$begingroup$
I don't know if i used the right word because i tranlated the problem from Greek. But it basically asks you to prove that C($mathbb{R}$),$d_0$) is complete or not.
$endgroup$
– argiriskar
Oct 11 '18 at 11:36
$begingroup$
This post answers your question. In order for a metric space to be complete, every Cauchy sequence must converges (with respect to the metric defined on the space). Every Cauchy sequence in your space must be constant for $n$ large enough, so it converges.
$endgroup$
– nicomezi
Oct 11 '18 at 11:42
add a comment |
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1 Answer
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active
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1 Answer
1
active
oldest
votes
active
oldest
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active
oldest
votes
$begingroup$
If you endow any set with that discrete metric, it becomes a complete metric space, because any Cauchy sequence is constant then, if $n$ is large enough.
answered Oct 11 '18 at 11:31
José Carlos SantosJosé Carlos Santos
153k22123225
$endgroup$
$begingroup$
I don't know if i used the right word because i tranlated the problem from Greek. But it basically asks you to prove that C($mathbb{R}$),$d_0$) is complete or not.
$endgroup$
– argiriskar
Oct 11 '18 at 11:36
$begingroup$
This post answers your question. In order for a metric space to be complete, every Cauchy sequence must converges (with respect to the metric defined on the space). Every Cauchy sequence in your space must be constant for $n$ large enough, so it converges.
$endgroup$
– nicomezi
Oct 11 '18 at 11:42
add a comment |
$begingroup$
If you endow any set with that discrete metric, it becomes a complete metric space, because any Cauchy sequence is constant then, if $n$ is large enough.
answered Oct 11 '18 at 11:31
José Carlos SantosJosé Carlos Santos
153k22123225
$endgroup$
$begingroup$
I don't know if i used the right word because i tranlated the problem from Greek. But it basically asks you to prove that C($mathbb{R}$),$d_0$) is complete or not.
$endgroup$
– argiriskar
Oct 11 '18 at 11:36
$begingroup$
This post answers your question. In order for a metric space to be complete, every Cauchy sequence must converges (with respect to the metric defined on the space). Every Cauchy sequence in your space must be constant for $n$ large enough, so it converges.
$endgroup$
– nicomezi
Oct 11 '18 at 11:42
add a comment |
$begingroup$
If you endow any set with that discrete metric, it becomes a complete metric space, because any Cauchy sequence is constant then, if $n$ is large enough.
answered Oct 11 '18 at 11:31
José Carlos SantosJosé Carlos Santos
153k22123225
$endgroup$
If you endow any set with that discrete metric, it becomes a complete metric space, because any Cauchy sequence is constant then, if $n$ is large enough.
answered Oct 11 '18 at 11:31
José Carlos SantosJosé Carlos Santos
153k22123225
edited Jan 7 at 13:48
answered Oct 11 '18 at 11:31
José Carlos SantosJosé Carlos Santos
153k22123225
answered Oct 11 '18 at 11:31
José Carlos SantosJosé Carlos Santos
153k22123225
answered Oct 11 '18 at 11:31
José Carlos SantosJosé Carlos Santos
153k22123225
153k22123225
$begingroup$
I don't know if i used the right word because i tranlated the problem from Greek. But it basically asks you to prove that C($mathbb{R}$),$d_0$) is complete or not.
$endgroup$
– argiriskar
Oct 11 '18 at 11:36
$begingroup$
This post answers your question. In order for a metric space to be complete, every Cauchy sequence must converges (with respect to the metric defined on the space). Every Cauchy sequence in your space must be constant for $n$ large enough, so it converges.
$endgroup$
– nicomezi
Oct 11 '18 at 11:42
add a comment |
$begingroup$
I don't know if i used the right word because i tranlated the problem from Greek. But it basically asks you to prove that C($mathbb{R}$),$d_0$) is complete or not.
$endgroup$
– argiriskar
Oct 11 '18 at 11:36
$begingroup$
This post answers your question. In order for a metric space to be complete, every Cauchy sequence must converges (with respect to the metric defined on the space). Every Cauchy sequence in your space must be constant for $n$ large enough, so it converges.
$endgroup$
– nicomezi
Oct 11 '18 at 11:42
$begingroup$
I don't know if i used the right word because i tranlated the problem from Greek. But it basically asks you to prove that C($mathbb{R}$),$d_0$) is complete or not.
$endgroup$
– argiriskar
Oct 11 '18 at 11:36
$begingroup$
I don't know if i used the right word because i tranlated the problem from Greek. But it basically asks you to prove that C($mathbb{R}$),$d_0$) is complete or not.
$endgroup$
– argiriskar
Oct 11 '18 at 11:36
$begingroup$
This post answers your question. In order for a metric space to be complete, every Cauchy sequence must converges (with respect to the metric defined on the space). Every Cauchy sequence in your space must be constant for $n$ large enough, so it converges.
$endgroup$
– nicomezi
Oct 11 '18 at 11:42
$begingroup$
This post answers your question. In order for a metric space to be complete, every Cauchy sequence must converges (with respect to the metric defined on the space). Every Cauchy sequence in your space must be constant for $n$ large enough, so it converges.
$endgroup$
– nicomezi
Oct 11 '18 at 11:42
add a comment |
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