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What is meant by “infinitely often†in this problem on Borel sets?
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As the title suggests, I am having some trouble understanding an exercise regarding Borel sets. In particular I am trying to
Show that the set of real numbers that have a decimal expansion with
the digit $5$ appearing infinitely often is a Borel set.
My question here is what is meant by the phrase "infinitely often" in this case? I have never seen this phrase before in a formal context. This attempt of mine is horrific so don't butcher me, but I tried starting out in the following way:
Let $mathcalA_5$ denote the set of real numbers that have a decimal expansion with the digit $5$ appearing infinitely often, i.e. every $ainmathcalA_5$ can be written as $$a=ldots+ 5cdot10^n+5cdot10^n-1+ldots+5cdot 10 +5+5cdot10^-1+ldots 5cdot10^-m-1+5cdot10^-m+ldots$$
Am I interpreting "infinitely often" correct here? If not, what is it trying to say?
real-analysis measure-theory borel-sets
asked 1 hour ago
Thy Art is Math
39219
add a comment |Â
up vote
2
down vote
favorite
As the title suggests, I am having some trouble understanding an exercise regarding Borel sets. In particular I am trying to
Show that the set of real numbers that have a decimal expansion with
the digit $5$ appearing infinitely often is a Borel set.
My question here is what is meant by the phrase "infinitely often" in this case? I have never seen this phrase before in a formal context. This attempt of mine is horrific so don't butcher me, but I tried starting out in the following way:
Let $mathcalA_5$ denote the set of real numbers that have a decimal expansion with the digit $5$ appearing infinitely often, i.e. every $ainmathcalA_5$ can be written as $$a=ldots+ 5cdot10^n+5cdot10^n-1+ldots+5cdot 10 +5+5cdot10^-1+ldots 5cdot10^-m-1+5cdot10^-m+ldots$$
Am I interpreting "infinitely often" correct here? If not, what is it trying to say?
real-analysis measure-theory borel-sets
asked 1 hour ago
Thy Art is Math
39219
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
As the title suggests, I am having some trouble understanding an exercise regarding Borel sets. In particular I am trying to
Show that the set of real numbers that have a decimal expansion with
the digit $5$ appearing infinitely often is a Borel set.
My question here is what is meant by the phrase "infinitely often" in this case? I have never seen this phrase before in a formal context. This attempt of mine is horrific so don't butcher me, but I tried starting out in the following way:
Let $mathcalA_5$ denote the set of real numbers that have a decimal expansion with the digit $5$ appearing infinitely often, i.e. every $ainmathcalA_5$ can be written as $$a=ldots+ 5cdot10^n+5cdot10^n-1+ldots+5cdot 10 +5+5cdot10^-1+ldots 5cdot10^-m-1+5cdot10^-m+ldots$$
Am I interpreting "infinitely often" correct here? If not, what is it trying to say?
real-analysis measure-theory borel-sets
asked 1 hour ago
Thy Art is Math
39219
As the title suggests, I am having some trouble understanding an exercise regarding Borel sets. In particular I am trying to
Show that the set of real numbers that have a decimal expansion with
the digit $5$ appearing infinitely often is a Borel set.
My question here is what is meant by the phrase "infinitely often" in this case? I have never seen this phrase before in a formal context. This attempt of mine is horrific so don't butcher me, but I tried starting out in the following way:
Let $mathcalA_5$ denote the set of real numbers that have a decimal expansion with the digit $5$ appearing infinitely often, i.e. every $ainmathcalA_5$ can be written as $$a=ldots+ 5cdot10^n+5cdot10^n-1+ldots+5cdot 10 +5+5cdot10^-1+ldots 5cdot10^-m-1+5cdot10^-m+ldots$$
Am I interpreting "infinitely often" correct here? If not, what is it trying to say?
real-analysis measure-theory borel-sets
real-analysis measure-theory borel-sets
asked 1 hour ago
Thy Art is Math
39219
asked 1 hour ago
Thy Art is Math
39219
asked 1 hour ago
Thy Art is Math
39219
asked 1 hour ago
Thy Art is Math
39219
asked 1 hour ago
Thy Art is Math
39219
39219
add a comment |Â
add a comment |Â
2 Answers
2
active
oldest
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up vote
3
down vote
accepted
The phrase means just this; if $$a_k 10^k + a_k-1 10^k-1 + dots + a_0 + a_-1 10^-1 + a_-2 10^-2 + cdots$$
is the decimal expansion of a real number, then the set
$$
n : a_n = 5, -infty < n leq k
$$
is an infinite set.
Pulled up from the comments below: to elaborate, the given condition means that there is some infinite subsequence of $5$'s in the sequence $( a_n )$.
Take care that it does not mean that all $a_n$'s are $5$, or that all $a_n$'s after a certain point are $5$. For example, you could have that $a_n = 5$ whenever $n$ is even and $a_n = 7$ whenever $n$ is odd. This is a number whose decimal expansion contains infinitely many $5$'s, but it also contains infinitely many non-$5$'s.
answered 1 hour ago
Brahadeesh
4,23231550
So, in other words, there is some infinite "string" of $5$'s in each number in this set of real numbers?
– Thy Art is Math
58 mins ago
1
@ThyArtisMath Yes, there is some infinite string in the sequence $( a_n )$. Take care that it does not mean that all $a_n$'s are $5$, or that all $a_n$'s after a certain point are $5$. For example, you could have that $a_n = 5$ whenever $n$ is even and $a_n = 7$ whenever $n$ is odd. This is a number whose decimal expansion contains infinitely many $5$'s, but it also contains infinitely many non-$5$'s.
– Brahadeesh
55 mins ago
Got it! Thank you for the example -- that makes sense to me now!
– Thy Art is Math
53 mins ago
1
@ThyArtisMath Glad to be of help. :)
– Brahadeesh
50 mins ago
add a comment |Â
up vote
3
down vote
"Infinitely often" means "infinitely many times". In other words, of all of the digits in the decimal expansion of your number, infinitely many of them are $5$. It doesn't mean that all of them are $5$, or even that "most" of them are $5$ in any sense, just that there are infinitely many $5$s.
answered 1 hour ago
Eric Wofsey
167k12196310
Ah, that makes sense. Thank you for the clarification. Would you happen to have any suggestions for a way in which I can explicitly write such a number mathematically? Or do you think that writing that out would be unhelpful in this case?
– Thy Art is Math
1 hour ago
1
I doubt such a representation would be helpful. That would be essentially writing your set as the image of some function, but that is not likely to be helpful for showing it is Borel since an image of a Borel set under a Borel function need not be Borel
– Eric Wofsey
59 mins ago
Ah ok, I did not know that. Thanks for your feedback!
– Thy Art is Math
54 mins ago
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
The phrase means just this; if $$a_k 10^k + a_k-1 10^k-1 + dots + a_0 + a_-1 10^-1 + a_-2 10^-2 + cdots$$
is the decimal expansion of a real number, then the set
$$
n : a_n = 5, -infty < n leq k
$$
is an infinite set.
Pulled up from the comments below: to elaborate, the given condition means that there is some infinite subsequence of $5$'s in the sequence $( a_n )$.
Take care that it does not mean that all $a_n$'s are $5$, or that all $a_n$'s after a certain point are $5$. For example, you could have that $a_n = 5$ whenever $n$ is even and $a_n = 7$ whenever $n$ is odd. This is a number whose decimal expansion contains infinitely many $5$'s, but it also contains infinitely many non-$5$'s.
answered 1 hour ago
Brahadeesh
4,23231550
So, in other words, there is some infinite "string" of $5$'s in each number in this set of real numbers?
– Thy Art is Math
58 mins ago
1
@ThyArtisMath Yes, there is some infinite string in the sequence $( a_n )$. Take care that it does not mean that all $a_n$'s are $5$, or that all $a_n$'s after a certain point are $5$. For example, you could have that $a_n = 5$ whenever $n$ is even and $a_n = 7$ whenever $n$ is odd. This is a number whose decimal expansion contains infinitely many $5$'s, but it also contains infinitely many non-$5$'s.
– Brahadeesh
55 mins ago
Got it! Thank you for the example -- that makes sense to me now!
– Thy Art is Math
53 mins ago
1
@ThyArtisMath Glad to be of help. :)
– Brahadeesh
50 mins ago
add a comment |Â
up vote
3
down vote
accepted
The phrase means just this; if $$a_k 10^k + a_k-1 10^k-1 + dots + a_0 + a_-1 10^-1 + a_-2 10^-2 + cdots$$
is the decimal expansion of a real number, then the set
$$
n : a_n = 5, -infty < n leq k
$$
is an infinite set.
Pulled up from the comments below: to elaborate, the given condition means that there is some infinite subsequence of $5$'s in the sequence $( a_n )$.
Take care that it does not mean that all $a_n$'s are $5$, or that all $a_n$'s after a certain point are $5$. For example, you could have that $a_n = 5$ whenever $n$ is even and $a_n = 7$ whenever $n$ is odd. This is a number whose decimal expansion contains infinitely many $5$'s, but it also contains infinitely many non-$5$'s.
answered 1 hour ago
Brahadeesh
4,23231550
So, in other words, there is some infinite "string" of $5$'s in each number in this set of real numbers?
– Thy Art is Math
58 mins ago
1
@ThyArtisMath Yes, there is some infinite string in the sequence $( a_n )$. Take care that it does not mean that all $a_n$'s are $5$, or that all $a_n$'s after a certain point are $5$. For example, you could have that $a_n = 5$ whenever $n$ is even and $a_n = 7$ whenever $n$ is odd. This is a number whose decimal expansion contains infinitely many $5$'s, but it also contains infinitely many non-$5$'s.
– Brahadeesh
55 mins ago
Got it! Thank you for the example -- that makes sense to me now!
– Thy Art is Math
53 mins ago
1
@ThyArtisMath Glad to be of help. :)
– Brahadeesh
50 mins ago
add a comment |Â
up vote
3
down vote
accepted
up vote
3
down vote
accepted
The phrase means just this; if $$a_k 10^k + a_k-1 10^k-1 + dots + a_0 + a_-1 10^-1 + a_-2 10^-2 + cdots$$
is the decimal expansion of a real number, then the set
$$
n : a_n = 5, -infty < n leq k
$$
is an infinite set.
Pulled up from the comments below: to elaborate, the given condition means that there is some infinite subsequence of $5$'s in the sequence $( a_n )$.
Take care that it does not mean that all $a_n$'s are $5$, or that all $a_n$'s after a certain point are $5$. For example, you could have that $a_n = 5$ whenever $n$ is even and $a_n = 7$ whenever $n$ is odd. This is a number whose decimal expansion contains infinitely many $5$'s, but it also contains infinitely many non-$5$'s.
answered 1 hour ago
Brahadeesh
4,23231550
The phrase means just this; if $$a_k 10^k + a_k-1 10^k-1 + dots + a_0 + a_-1 10^-1 + a_-2 10^-2 + cdots$$
is the decimal expansion of a real number, then the set
$$
n : a_n = 5, -infty < n leq k
$$
is an infinite set.
Pulled up from the comments below: to elaborate, the given condition means that there is some infinite subsequence of $5$'s in the sequence $( a_n )$.
Take care that it does not mean that all $a_n$'s are $5$, or that all $a_n$'s after a certain point are $5$. For example, you could have that $a_n = 5$ whenever $n$ is even and $a_n = 7$ whenever $n$ is odd. This is a number whose decimal expansion contains infinitely many $5$'s, but it also contains infinitely many non-$5$'s.
answered 1 hour ago
Brahadeesh
4,23231550
edited 50 mins ago
answered 1 hour ago
Brahadeesh
4,23231550
answered 1 hour ago
Brahadeesh
4,23231550
answered 1 hour ago
Brahadeesh
4,23231550
4,23231550
So, in other words, there is some infinite "string" of $5$'s in each number in this set of real numbers?
– Thy Art is Math
58 mins ago
1
@ThyArtisMath Yes, there is some infinite string in the sequence $( a_n )$. Take care that it does not mean that all $a_n$'s are $5$, or that all $a_n$'s after a certain point are $5$. For example, you could have that $a_n = 5$ whenever $n$ is even and $a_n = 7$ whenever $n$ is odd. This is a number whose decimal expansion contains infinitely many $5$'s, but it also contains infinitely many non-$5$'s.
– Brahadeesh
55 mins ago
Got it! Thank you for the example -- that makes sense to me now!
– Thy Art is Math
53 mins ago
1
@ThyArtisMath Glad to be of help. :)
– Brahadeesh
50 mins ago
add a comment |Â
So, in other words, there is some infinite "string" of $5$'s in each number in this set of real numbers?
– Thy Art is Math
58 mins ago
1
@ThyArtisMath Yes, there is some infinite string in the sequence $( a_n )$. Take care that it does not mean that all $a_n$'s are $5$, or that all $a_n$'s after a certain point are $5$. For example, you could have that $a_n = 5$ whenever $n$ is even and $a_n = 7$ whenever $n$ is odd. This is a number whose decimal expansion contains infinitely many $5$'s, but it also contains infinitely many non-$5$'s.
– Brahadeesh
55 mins ago
Got it! Thank you for the example -- that makes sense to me now!
– Thy Art is Math
53 mins ago
1
@ThyArtisMath Glad to be of help. :)
– Brahadeesh
50 mins ago
So, in other words, there is some infinite "string" of $5$'s in each number in this set of real numbers?
– Thy Art is Math
58 mins ago
So, in other words, there is some infinite "string" of $5$'s in each number in this set of real numbers?
– Thy Art is Math
58 mins ago
1
1
@ThyArtisMath Yes, there is some infinite string in the sequence $( a_n )$. Take care that it does not mean that all $a_n$'s are $5$, or that all $a_n$'s after a certain point are $5$. For example, you could have that $a_n = 5$ whenever $n$ is even and $a_n = 7$ whenever $n$ is odd. This is a number whose decimal expansion contains infinitely many $5$'s, but it also contains infinitely many non-$5$'s.
– Brahadeesh
55 mins ago
@ThyArtisMath Yes, there is some infinite string in the sequence $( a_n )$. Take care that it does not mean that all $a_n$'s are $5$, or that all $a_n$'s after a certain point are $5$. For example, you could have that $a_n = 5$ whenever $n$ is even and $a_n = 7$ whenever $n$ is odd. This is a number whose decimal expansion contains infinitely many $5$'s, but it also contains infinitely many non-$5$'s.
– Brahadeesh
55 mins ago
Got it! Thank you for the example -- that makes sense to me now!
– Thy Art is Math
53 mins ago
Got it! Thank you for the example -- that makes sense to me now!
– Thy Art is Math
53 mins ago
1
1
@ThyArtisMath Glad to be of help. :)
– Brahadeesh
50 mins ago
@ThyArtisMath Glad to be of help. :)
– Brahadeesh
50 mins ago
add a comment |Â
up vote
3
down vote
"Infinitely often" means "infinitely many times". In other words, of all of the digits in the decimal expansion of your number, infinitely many of them are $5$. It doesn't mean that all of them are $5$, or even that "most" of them are $5$ in any sense, just that there are infinitely many $5$s.
answered 1 hour ago
Eric Wofsey
167k12196310
Ah, that makes sense. Thank you for the clarification. Would you happen to have any suggestions for a way in which I can explicitly write such a number mathematically? Or do you think that writing that out would be unhelpful in this case?
– Thy Art is Math
1 hour ago
1
I doubt such a representation would be helpful. That would be essentially writing your set as the image of some function, but that is not likely to be helpful for showing it is Borel since an image of a Borel set under a Borel function need not be Borel
– Eric Wofsey
59 mins ago
Ah ok, I did not know that. Thanks for your feedback!
– Thy Art is Math
54 mins ago
add a comment |Â
up vote
3
down vote
"Infinitely often" means "infinitely many times". In other words, of all of the digits in the decimal expansion of your number, infinitely many of them are $5$. It doesn't mean that all of them are $5$, or even that "most" of them are $5$ in any sense, just that there are infinitely many $5$s.
answered 1 hour ago
Eric Wofsey
167k12196310
Ah, that makes sense. Thank you for the clarification. Would you happen to have any suggestions for a way in which I can explicitly write such a number mathematically? Or do you think that writing that out would be unhelpful in this case?
– Thy Art is Math
1 hour ago
1
I doubt such a representation would be helpful. That would be essentially writing your set as the image of some function, but that is not likely to be helpful for showing it is Borel since an image of a Borel set under a Borel function need not be Borel
– Eric Wofsey
59 mins ago
Ah ok, I did not know that. Thanks for your feedback!
– Thy Art is Math
54 mins ago
add a comment |Â
up vote
3
down vote
up vote
3
down vote
"Infinitely often" means "infinitely many times". In other words, of all of the digits in the decimal expansion of your number, infinitely many of them are $5$. It doesn't mean that all of them are $5$, or even that "most" of them are $5$ in any sense, just that there are infinitely many $5$s.
answered 1 hour ago
Eric Wofsey
167k12196310
"Infinitely often" means "infinitely many times". In other words, of all of the digits in the decimal expansion of your number, infinitely many of them are $5$. It doesn't mean that all of them are $5$, or even that "most" of them are $5$ in any sense, just that there are infinitely many $5$s.
answered 1 hour ago
Eric Wofsey
167k12196310
answered 1 hour ago
Eric Wofsey
167k12196310
answered 1 hour ago
Eric Wofsey
167k12196310
answered 1 hour ago
Eric Wofsey
167k12196310
167k12196310
Ah, that makes sense. Thank you for the clarification. Would you happen to have any suggestions for a way in which I can explicitly write such a number mathematically? Or do you think that writing that out would be unhelpful in this case?
– Thy Art is Math
1 hour ago
1
I doubt such a representation would be helpful. That would be essentially writing your set as the image of some function, but that is not likely to be helpful for showing it is Borel since an image of a Borel set under a Borel function need not be Borel
– Eric Wofsey
59 mins ago
Ah ok, I did not know that. Thanks for your feedback!
– Thy Art is Math
54 mins ago
add a comment |Â
Ah, that makes sense. Thank you for the clarification. Would you happen to have any suggestions for a way in which I can explicitly write such a number mathematically? Or do you think that writing that out would be unhelpful in this case?
– Thy Art is Math
1 hour ago
1
I doubt such a representation would be helpful. That would be essentially writing your set as the image of some function, but that is not likely to be helpful for showing it is Borel since an image of a Borel set under a Borel function need not be Borel
– Eric Wofsey
59 mins ago
Ah ok, I did not know that. Thanks for your feedback!
– Thy Art is Math
54 mins ago
Ah, that makes sense. Thank you for the clarification. Would you happen to have any suggestions for a way in which I can explicitly write such a number mathematically? Or do you think that writing that out would be unhelpful in this case?
– Thy Art is Math
1 hour ago
Ah, that makes sense. Thank you for the clarification. Would you happen to have any suggestions for a way in which I can explicitly write such a number mathematically? Or do you think that writing that out would be unhelpful in this case?
– Thy Art is Math
1 hour ago
1
1
I doubt such a representation would be helpful. That would be essentially writing your set as the image of some function, but that is not likely to be helpful for showing it is Borel since an image of a Borel set under a Borel function need not be Borel
– Eric Wofsey
59 mins ago
I doubt such a representation would be helpful. That would be essentially writing your set as the image of some function, but that is not likely to be helpful for showing it is Borel since an image of a Borel set under a Borel function need not be Borel
– Eric Wofsey
59 mins ago
Ah ok, I did not know that. Thanks for your feedback!
– Thy Art is Math
54 mins ago
Ah ok, I did not know that. Thanks for your feedback!
– Thy Art is Math
54 mins ago
add a comment |Â
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