Any sequence in metric space has lim supremum
$begingroup$
Does any sequence in any metric space have a limit supremum and a limit infimum?
If the sequence is growing, it would be $infty$ if unbounded or a finite number if bounded.
If the sequence is decreasing, then it would be -$infty$ or some finite number if bounded.
If the sequence oscillating, it should still have $+-infty$ as an upper and lower bound. I guess it could also have $infty^n$, but I am not sure about that.
If the sequence is random, but we know all of the elements in the sequence, shouldn't it still have bounds?
If it is not true that all sequences have lim supremum/infimum, then can someone provide an example or explanation please?
real-analysis
asked Dec 10 '18 at 18:44
FrankFrank
17610
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show 2 more comments
$begingroup$
Does any sequence in any metric space have a limit supremum and a limit infimum?
If the sequence is growing, it would be $infty$ if unbounded or a finite number if bounded.
If the sequence is decreasing, then it would be -$infty$ or some finite number if bounded.
If the sequence oscillating, it should still have $+-infty$ as an upper and lower bound. I guess it could also have $infty^n$, but I am not sure about that.
If the sequence is random, but we know all of the elements in the sequence, shouldn't it still have bounds?
If it is not true that all sequences have lim supremum/infimum, then can someone provide an example or explanation please?
real-analysis
asked Dec 10 '18 at 18:44
FrankFrank
17610
$endgroup$
1
$begingroup$
Asking this about "any metric space" makes no sense. What do "decreasing" or "oscillating" mean for a set of points in the plane?
$endgroup$
– Ethan Bolker
Dec 10 '18 at 18:49
$begingroup$
I'm asking about an arbitrary metric space. Why would it make no sense? Either there is an example of a metric space with out a limit supremum defined, or there isn't?
$endgroup$
– Frank
Dec 10 '18 at 19:22
2
$begingroup$
To ask about "growing" or "lim sup" the underlying space must have a linear order. Most metric spaces don't. The plane is an example where "lim sup" is not defined. so you can't ask whether or not a sequence has one.
$endgroup$
– Ethan Bolker
Dec 10 '18 at 19:30
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Not following. What is an example of a sequence of numbers in a plane that does not have a limit supremum, if you can consider infinity as a limit supremum? The plane does not have a linear order? I don't get that.
$endgroup$
– Frank
Dec 10 '18 at 20:15
$begingroup$
There is no "infinity" in the plane for a sequence of points to have as a lim sup. Perhaps you can learn more from wikipedia: en.wikipedia.org/wiki/…
$endgroup$
– Ethan Bolker
Dec 10 '18 at 20:54
|
show 2 more comments
$begingroup$
Does any sequence in any metric space have a limit supremum and a limit infimum?
If the sequence is growing, it would be $infty$ if unbounded or a finite number if bounded.
If the sequence is decreasing, then it would be -$infty$ or some finite number if bounded.
If the sequence oscillating, it should still have $+-infty$ as an upper and lower bound. I guess it could also have $infty^n$, but I am not sure about that.
If the sequence is random, but we know all of the elements in the sequence, shouldn't it still have bounds?
If it is not true that all sequences have lim supremum/infimum, then can someone provide an example or explanation please?
real-analysis
asked Dec 10 '18 at 18:44
FrankFrank
17610
$endgroup$
Does any sequence in any metric space have a limit supremum and a limit infimum?
If the sequence is growing, it would be $infty$ if unbounded or a finite number if bounded.
If the sequence is decreasing, then it would be -$infty$ or some finite number if bounded.
If the sequence oscillating, it should still have $+-infty$ as an upper and lower bound. I guess it could also have $infty^n$, but I am not sure about that.
If the sequence is random, but we know all of the elements in the sequence, shouldn't it still have bounds?
If it is not true that all sequences have lim supremum/infimum, then can someone provide an example or explanation please?
real-analysis
real-analysis
asked Dec 10 '18 at 18:44
FrankFrank
17610
asked Dec 10 '18 at 18:44
FrankFrank
17610
asked Dec 10 '18 at 18:44
FrankFrank
17610
asked Dec 10 '18 at 18:44
FrankFrank
17610
asked Dec 10 '18 at 18:44
FrankFrank
17610
17610
1
$begingroup$
Asking this about "any metric space" makes no sense. What do "decreasing" or "oscillating" mean for a set of points in the plane?
$endgroup$
– Ethan Bolker
Dec 10 '18 at 18:49
$begingroup$
I'm asking about an arbitrary metric space. Why would it make no sense? Either there is an example of a metric space with out a limit supremum defined, or there isn't?
$endgroup$
– Frank
Dec 10 '18 at 19:22
2
$begingroup$
To ask about "growing" or "lim sup" the underlying space must have a linear order. Most metric spaces don't. The plane is an example where "lim sup" is not defined. so you can't ask whether or not a sequence has one.
$endgroup$
– Ethan Bolker
Dec 10 '18 at 19:30
$begingroup$
Not following. What is an example of a sequence of numbers in a plane that does not have a limit supremum, if you can consider infinity as a limit supremum? The plane does not have a linear order? I don't get that.
$endgroup$
– Frank
Dec 10 '18 at 20:15
$begingroup$
There is no "infinity" in the plane for a sequence of points to have as a lim sup. Perhaps you can learn more from wikipedia: en.wikipedia.org/wiki/…
$endgroup$
– Ethan Bolker
Dec 10 '18 at 20:54
|
show 2 more comments
1
$begingroup$
Asking this about "any metric space" makes no sense. What do "decreasing" or "oscillating" mean for a set of points in the plane?
$endgroup$
– Ethan Bolker
Dec 10 '18 at 18:49
$begingroup$
I'm asking about an arbitrary metric space. Why would it make no sense? Either there is an example of a metric space with out a limit supremum defined, or there isn't?
$endgroup$
– Frank
Dec 10 '18 at 19:22
2
$begingroup$
To ask about "growing" or "lim sup" the underlying space must have a linear order. Most metric spaces don't. The plane is an example where "lim sup" is not defined. so you can't ask whether or not a sequence has one.
$endgroup$
– Ethan Bolker
Dec 10 '18 at 19:30
$begingroup$
Not following. What is an example of a sequence of numbers in a plane that does not have a limit supremum, if you can consider infinity as a limit supremum? The plane does not have a linear order? I don't get that.
$endgroup$
– Frank
Dec 10 '18 at 20:15
$begingroup$
There is no "infinity" in the plane for a sequence of points to have as a lim sup. Perhaps you can learn more from wikipedia: en.wikipedia.org/wiki/…
$endgroup$
– Ethan Bolker
Dec 10 '18 at 20:54
1
1
$begingroup$
Asking this about "any metric space" makes no sense. What do "decreasing" or "oscillating" mean for a set of points in the plane?
$endgroup$
– Ethan Bolker
Dec 10 '18 at 18:49
$begingroup$
Asking this about "any metric space" makes no sense. What do "decreasing" or "oscillating" mean for a set of points in the plane?
$endgroup$
– Ethan Bolker
Dec 10 '18 at 18:49
$begingroup$
I'm asking about an arbitrary metric space. Why would it make no sense? Either there is an example of a metric space with out a limit supremum defined, or there isn't?
$endgroup$
– Frank
Dec 10 '18 at 19:22
$begingroup$
I'm asking about an arbitrary metric space. Why would it make no sense? Either there is an example of a metric space with out a limit supremum defined, or there isn't?
$endgroup$
– Frank
Dec 10 '18 at 19:22
2
2
$begingroup$
To ask about "growing" or "lim sup" the underlying space must have a linear order. Most metric spaces don't. The plane is an example where "lim sup" is not defined. so you can't ask whether or not a sequence has one.
$endgroup$
– Ethan Bolker
Dec 10 '18 at 19:30
$begingroup$
To ask about "growing" or "lim sup" the underlying space must have a linear order. Most metric spaces don't. The plane is an example where "lim sup" is not defined. so you can't ask whether or not a sequence has one.
$endgroup$
– Ethan Bolker
Dec 10 '18 at 19:30
$begingroup$
Not following. What is an example of a sequence of numbers in a plane that does not have a limit supremum, if you can consider infinity as a limit supremum? The plane does not have a linear order? I don't get that.
$endgroup$
– Frank
Dec 10 '18 at 20:15
$begingroup$
Not following. What is an example of a sequence of numbers in a plane that does not have a limit supremum, if you can consider infinity as a limit supremum? The plane does not have a linear order? I don't get that.
$endgroup$
– Frank
Dec 10 '18 at 20:15
$begingroup$
There is no "infinity" in the plane for a sequence of points to have as a lim sup. Perhaps you can learn more from wikipedia: en.wikipedia.org/wiki/…
$endgroup$
– Ethan Bolker
Dec 10 '18 at 20:54
$begingroup$
There is no "infinity" in the plane for a sequence of points to have as a lim sup. Perhaps you can learn more from wikipedia: en.wikipedia.org/wiki/…
$endgroup$
– Ethan Bolker
Dec 10 '18 at 20:54
|
show 2 more comments
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$begingroup$
Asking this about "any metric space" makes no sense. What do "decreasing" or "oscillating" mean for a set of points in the plane?
$endgroup$
– Ethan Bolker
Dec 10 '18 at 18:49
$begingroup$
I'm asking about an arbitrary metric space. Why would it make no sense? Either there is an example of a metric space with out a limit supremum defined, or there isn't?
$endgroup$
– Frank
Dec 10 '18 at 19:22
2
$begingroup$
To ask about "growing" or "lim sup" the underlying space must have a linear order. Most metric spaces don't. The plane is an example where "lim sup" is not defined. so you can't ask whether or not a sequence has one.
$endgroup$
– Ethan Bolker
Dec 10 '18 at 19:30
$begingroup$
Not following. What is an example of a sequence of numbers in a plane that does not have a limit supremum, if you can consider infinity as a limit supremum? The plane does not have a linear order? I don't get that.
$endgroup$
– Frank
Dec 10 '18 at 20:15
$begingroup$
There is no "infinity" in the plane for a sequence of points to have as a lim sup. Perhaps you can learn more from wikipedia: en.wikipedia.org/wiki/…
$endgroup$
– Ethan Bolker
Dec 10 '18 at 20:54