Isolated points and limit points
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I’m reading a complex analysis textbook, and they do a brief review on real analysis. I’ve attached a screenshot of the page, and highlighted the statement in question. Here is the statement:
“Clearly, for the sequence $lbrace 1,1,2,1,2,3,1,2,3,4,...rbrace$, each natural number is a limit point.”
My question is: since the sequence above is made up of only natural numbers, aren’t each of the numbers in the sequence isolated points, and hence, can’t be limit points? Not every neighbourhood of the point 12, say, contains infinitely many terms of the the sequence $lbrace 1,1,2,1,2,3,1,2,3,4,...rbrace$. Am I missing something? 
real-analysis
asked Dec 14 '18 at 20:39
Live Free or π HardLive Free or π Hard
481213
$endgroup$
add a comment |
$begingroup$
I’m reading a complex analysis textbook, and they do a brief review on real analysis. I’ve attached a screenshot of the page, and highlighted the statement in question. Here is the statement:
“Clearly, for the sequence $lbrace 1,1,2,1,2,3,1,2,3,4,...rbrace$, each natural number is a limit point.”
My question is: since the sequence above is made up of only natural numbers, aren’t each of the numbers in the sequence isolated points, and hence, can’t be limit points? Not every neighbourhood of the point 12, say, contains infinitely many terms of the the sequence $lbrace 1,1,2,1,2,3,1,2,3,4,...rbrace$. Am I missing something?
real-analysis
asked Dec 14 '18 at 20:39
Live Free or π HardLive Free or π Hard
481213
$endgroup$
1
$begingroup$
There is a distinction between the limit points of a sequence $(f(n))_{nin Bbb N}$ and the limit points of the set ${f(n): nin Bbb N}$.
$endgroup$
– DanielWainfleet
Dec 15 '18 at 6:01
add a comment |
$begingroup$
I’m reading a complex analysis textbook, and they do a brief review on real analysis. I’ve attached a screenshot of the page, and highlighted the statement in question. Here is the statement:
“Clearly, for the sequence $lbrace 1,1,2,1,2,3,1,2,3,4,...rbrace$, each natural number is a limit point.”
My question is: since the sequence above is made up of only natural numbers, aren’t each of the numbers in the sequence isolated points, and hence, can’t be limit points? Not every neighbourhood of the point 12, say, contains infinitely many terms of the the sequence $lbrace 1,1,2,1,2,3,1,2,3,4,...rbrace$. Am I missing something?
real-analysis
asked Dec 14 '18 at 20:39
Live Free or π HardLive Free or π Hard
481213
$endgroup$
I’m reading a complex analysis textbook, and they do a brief review on real analysis. I’ve attached a screenshot of the page, and highlighted the statement in question. Here is the statement:
“Clearly, for the sequence $lbrace 1,1,2,1,2,3,1,2,3,4,...rbrace$, each natural number is a limit point.”
My question is: since the sequence above is made up of only natural numbers, aren’t each of the numbers in the sequence isolated points, and hence, can’t be limit points? Not every neighbourhood of the point 12, say, contains infinitely many terms of the the sequence $lbrace 1,1,2,1,2,3,1,2,3,4,...rbrace$. Am I missing something?
real-analysis
real-analysis
asked Dec 14 '18 at 20:39
Live Free or π HardLive Free or π Hard
481213
asked Dec 14 '18 at 20:39
Live Free or π HardLive Free or π Hard
481213
asked Dec 14 '18 at 20:39
Live Free or π HardLive Free or π Hard
481213
asked Dec 14 '18 at 20:39
Live Free or π HardLive Free or π Hard
481213
asked Dec 14 '18 at 20:39
Live Free or π HardLive Free or π Hard
481213
481213
1
$begingroup$
There is a distinction between the limit points of a sequence $(f(n))_{nin Bbb N}$ and the limit points of the set ${f(n): nin Bbb N}$.
$endgroup$
– DanielWainfleet
Dec 15 '18 at 6:01
add a comment |
1
$begingroup$
There is a distinction between the limit points of a sequence $(f(n))_{nin Bbb N}$ and the limit points of the set ${f(n): nin Bbb N}$.
$endgroup$
– DanielWainfleet
Dec 15 '18 at 6:01
1
1
$begingroup$
There is a distinction between the limit points of a sequence $(f(n))_{nin Bbb N}$ and the limit points of the set ${f(n): nin Bbb N}$.
$endgroup$
– DanielWainfleet
Dec 15 '18 at 6:01
$begingroup$
There is a distinction between the limit points of a sequence $(f(n))_{nin Bbb N}$ and the limit points of the set ${f(n): nin Bbb N}$.
$endgroup$
– DanielWainfleet
Dec 15 '18 at 6:01
add a comment |
1 Answer
1
active
oldest
votes
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For every neighbourhood of $12$ there exist infinitely many indices where the sequence lies in this neighbourhood.
answered Dec 14 '18 at 20:43
SmileyCraftSmileyCraft
3,776519
$endgroup$
$begingroup$
@ SmileyCraft: thank you! Just to clarify: even if I take the neighbourhood around the point 12, that only includes the point 12, because the number 12 appears an infinite amount of times in my sequence, it’s a limit point. If the sequence was $lbrace 1,2,3,4,...rbrace$ then all the points in the sequence are isolated points, and hence, are not limit points. Correct?
$endgroup$
– Live Free or π Hard
Dec 14 '18 at 21:02
1
$begingroup$
Well the notion of a limit point has a different meaning depending on whether we consider a set or a sequence, but as far as I know, isolated points only make sense for sets. But if there is a notion of an isolated point of a sequence, then the canonical way to define this would coincide with your claim, so you probably understand it now.
$endgroup$
– SmileyCraft
Dec 14 '18 at 21:06
add a comment |
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1 Answer
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1 Answer
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active
oldest
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active
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votes
$begingroup$
For every neighbourhood of $12$ there exist infinitely many indices where the sequence lies in this neighbourhood.
answered Dec 14 '18 at 20:43
SmileyCraftSmileyCraft
3,776519
$endgroup$
$begingroup$
@ SmileyCraft: thank you! Just to clarify: even if I take the neighbourhood around the point 12, that only includes the point 12, because the number 12 appears an infinite amount of times in my sequence, it’s a limit point. If the sequence was $lbrace 1,2,3,4,...rbrace$ then all the points in the sequence are isolated points, and hence, are not limit points. Correct?
$endgroup$
– Live Free or π Hard
Dec 14 '18 at 21:02
1
$begingroup$
Well the notion of a limit point has a different meaning depending on whether we consider a set or a sequence, but as far as I know, isolated points only make sense for sets. But if there is a notion of an isolated point of a sequence, then the canonical way to define this would coincide with your claim, so you probably understand it now.
$endgroup$
– SmileyCraft
Dec 14 '18 at 21:06
add a comment |
$begingroup$
For every neighbourhood of $12$ there exist infinitely many indices where the sequence lies in this neighbourhood.
answered Dec 14 '18 at 20:43
SmileyCraftSmileyCraft
3,776519
$endgroup$
$begingroup$
@ SmileyCraft: thank you! Just to clarify: even if I take the neighbourhood around the point 12, that only includes the point 12, because the number 12 appears an infinite amount of times in my sequence, it’s a limit point. If the sequence was $lbrace 1,2,3,4,...rbrace$ then all the points in the sequence are isolated points, and hence, are not limit points. Correct?
$endgroup$
– Live Free or π Hard
Dec 14 '18 at 21:02
1
$begingroup$
Well the notion of a limit point has a different meaning depending on whether we consider a set or a sequence, but as far as I know, isolated points only make sense for sets. But if there is a notion of an isolated point of a sequence, then the canonical way to define this would coincide with your claim, so you probably understand it now.
$endgroup$
– SmileyCraft
Dec 14 '18 at 21:06
add a comment |
$begingroup$
For every neighbourhood of $12$ there exist infinitely many indices where the sequence lies in this neighbourhood.
answered Dec 14 '18 at 20:43
SmileyCraftSmileyCraft
3,776519
$endgroup$
For every neighbourhood of $12$ there exist infinitely many indices where the sequence lies in this neighbourhood.
answered Dec 14 '18 at 20:43
SmileyCraftSmileyCraft
3,776519
answered Dec 14 '18 at 20:43
SmileyCraftSmileyCraft
3,776519
answered Dec 14 '18 at 20:43
SmileyCraftSmileyCraft
3,776519
answered Dec 14 '18 at 20:43
SmileyCraftSmileyCraft
3,776519
3,776519
$begingroup$
@ SmileyCraft: thank you! Just to clarify: even if I take the neighbourhood around the point 12, that only includes the point 12, because the number 12 appears an infinite amount of times in my sequence, it’s a limit point. If the sequence was $lbrace 1,2,3,4,...rbrace$ then all the points in the sequence are isolated points, and hence, are not limit points. Correct?
$endgroup$
– Live Free or π Hard
Dec 14 '18 at 21:02
1
$begingroup$
Well the notion of a limit point has a different meaning depending on whether we consider a set or a sequence, but as far as I know, isolated points only make sense for sets. But if there is a notion of an isolated point of a sequence, then the canonical way to define this would coincide with your claim, so you probably understand it now.
$endgroup$
– SmileyCraft
Dec 14 '18 at 21:06
add a comment |
$begingroup$
@ SmileyCraft: thank you! Just to clarify: even if I take the neighbourhood around the point 12, that only includes the point 12, because the number 12 appears an infinite amount of times in my sequence, it’s a limit point. If the sequence was $lbrace 1,2,3,4,...rbrace$ then all the points in the sequence are isolated points, and hence, are not limit points. Correct?
$endgroup$
– Live Free or π Hard
Dec 14 '18 at 21:02
1
$begingroup$
Well the notion of a limit point has a different meaning depending on whether we consider a set or a sequence, but as far as I know, isolated points only make sense for sets. But if there is a notion of an isolated point of a sequence, then the canonical way to define this would coincide with your claim, so you probably understand it now.
$endgroup$
– SmileyCraft
Dec 14 '18 at 21:06
$begingroup$
@ SmileyCraft: thank you! Just to clarify: even if I take the neighbourhood around the point 12, that only includes the point 12, because the number 12 appears an infinite amount of times in my sequence, it’s a limit point. If the sequence was $lbrace 1,2,3,4,...rbrace$ then all the points in the sequence are isolated points, and hence, are not limit points. Correct?
$endgroup$
– Live Free or π Hard
Dec 14 '18 at 21:02
$begingroup$
@ SmileyCraft: thank you! Just to clarify: even if I take the neighbourhood around the point 12, that only includes the point 12, because the number 12 appears an infinite amount of times in my sequence, it’s a limit point. If the sequence was $lbrace 1,2,3,4,...rbrace$ then all the points in the sequence are isolated points, and hence, are not limit points. Correct?
$endgroup$
– Live Free or π Hard
Dec 14 '18 at 21:02
1
1
$begingroup$
Well the notion of a limit point has a different meaning depending on whether we consider a set or a sequence, but as far as I know, isolated points only make sense for sets. But if there is a notion of an isolated point of a sequence, then the canonical way to define this would coincide with your claim, so you probably understand it now.
$endgroup$
– SmileyCraft
Dec 14 '18 at 21:06
$begingroup$
Well the notion of a limit point has a different meaning depending on whether we consider a set or a sequence, but as far as I know, isolated points only make sense for sets. But if there is a notion of an isolated point of a sequence, then the canonical way to define this would coincide with your claim, so you probably understand it now.
$endgroup$
– SmileyCraft
Dec 14 '18 at 21:06
add a comment |
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$begingroup$
There is a distinction between the limit points of a sequence $(f(n))_{nin Bbb N}$ and the limit points of the set ${f(n): nin Bbb N}$.
$endgroup$
– DanielWainfleet
Dec 15 '18 at 6:01