Show that a certain set of positive real numbers must be finite or countable
Clash Royale CLAN TAG#URR8PPP
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Let $B$ be a set of positive real numbers with the property that
adding together any finite subset of elements from $B$ always gives a sum of $2$ or less. Show that $B$ must be finite or at most countable.
$B$ = $x in R:x>0$, $x_1,x_2...x_n in B$ such that $x_1+x_2+...+x_n le 2$.
Question: for any $a,b$ $(a,b)$~$R$, but $B$ is $(0,+infty)$ so why $B$ is not uncountable (taking as $a = 0$, and letting $b$->$infty$)?
And why for $B$ being countable doesn't contradict: for any $a,b$ $(a,b)$~$R$?
P.S. I read Showing a set is finite or countable and understood it.
real-analysis analysis
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show 6 more comments
up vote
3
down vote
favorite
Let $B$ be a set of positive real numbers with the property that
adding together any finite subset of elements from $B$ always gives a sum of $2$ or less. Show that $B$ must be finite or at most countable.
$B$ = $x in R:x>0$, $x_1,x_2...x_n in B$ such that $x_1+x_2+...+x_n le 2$.
Question: for any $a,b$ $(a,b)$~$R$, but $B$ is $(0,+infty)$ so why $B$ is not uncountable (taking as $a = 0$, and letting $b$->$infty$)?
And why for $B$ being countable doesn't contradict: for any $a,b$ $(a,b)$~$R$?
P.S. I read Showing a set is finite or countable and understood it.
real-analysis analysis
4
It sounds like you are confused about what the question is asking. It is not saying that B is the set of positive real numbers; it is a set of positive real numbers. This means that B is a subset of the set of all positive real numbers.
â Leonard Blackburn
Aug 8 at 17:23
2
No. B is some unspecified subset of the positive real numbers such that if you add up a finite number of the elements of B the sum is always less than 2. For example, B could be 0.1, 0.2, 0.5, 1.1 or B could be the infinite set 1, 1/2, 1/4, 1/8, 1/16, .... Note that B cannot be 1, 1.4, 1.8 and B cannot be 1, 1/2, 1/3, 1/4, ...
â Leonard Blackburn
Aug 8 at 17:47
2
B does not have to be an interval (0,a). In fact, it cannot be such an interval. Your job is to prove that B must be finite or countable. Intervals are uncountable.
â Leonard Blackburn
Aug 8 at 17:49
2
Well, I'm just trying to help clarify the question. I haven't given a solution. I think the hints and discussion below do that for you. But the point is you cannot assume anything about B except: 1. its elements are all positive real numbers and 2. if you take a finite number of elements from B and add them up then you get a sum less than two. Given any such B you have to prove that B is either finite or countable.
â Leonard Blackburn
Aug 8 at 17:59
1
Here is a proof that B cannot be the open interval (0,1). If it were then these numbers would be in B: 0.5, 0.51, 0.52, and 0.53. These numbers add up to more than 2. Do you see? You could do something similar to show that B cannot be (0,a) for any positive a. But you have to show more than that. You have to show B is finite or countable.
â Leonard Blackburn
Aug 8 at 18:00
 |Â
show 6 more comments
up vote
3
down vote
favorite
up vote
3
down vote
favorite
Let $B$ be a set of positive real numbers with the property that
adding together any finite subset of elements from $B$ always gives a sum of $2$ or less. Show that $B$ must be finite or at most countable.
$B$ = $x in R:x>0$, $x_1,x_2...x_n in B$ such that $x_1+x_2+...+x_n le 2$.
Question: for any $a,b$ $(a,b)$~$R$, but $B$ is $(0,+infty)$ so why $B$ is not uncountable (taking as $a = 0$, and letting $b$->$infty$)?
And why for $B$ being countable doesn't contradict: for any $a,b$ $(a,b)$~$R$?
P.S. I read Showing a set is finite or countable and understood it.
real-analysis analysis
Let $B$ be a set of positive real numbers with the property that
adding together any finite subset of elements from $B$ always gives a sum of $2$ or less. Show that $B$ must be finite or at most countable.
$B$ = $x in R:x>0$, $x_1,x_2...x_n in B$ such that $x_1+x_2+...+x_n le 2$.
Question: for any $a,b$ $(a,b)$~$R$, but $B$ is $(0,+infty)$ so why $B$ is not uncountable (taking as $a = 0$, and letting $b$->$infty$)?
And why for $B$ being countable doesn't contradict: for any $a,b$ $(a,b)$~$R$?
P.S. I read Showing a set is finite or countable and understood it.
real-analysis analysis
edited Aug 8 at 18:23
dmtri
768317
768317
asked Aug 8 at 16:57
Sargis Iskandaryan
49412
49412
4
It sounds like you are confused about what the question is asking. It is not saying that B is the set of positive real numbers; it is a set of positive real numbers. This means that B is a subset of the set of all positive real numbers.
â Leonard Blackburn
Aug 8 at 17:23
2
No. B is some unspecified subset of the positive real numbers such that if you add up a finite number of the elements of B the sum is always less than 2. For example, B could be 0.1, 0.2, 0.5, 1.1 or B could be the infinite set 1, 1/2, 1/4, 1/8, 1/16, .... Note that B cannot be 1, 1.4, 1.8 and B cannot be 1, 1/2, 1/3, 1/4, ...
â Leonard Blackburn
Aug 8 at 17:47
2
B does not have to be an interval (0,a). In fact, it cannot be such an interval. Your job is to prove that B must be finite or countable. Intervals are uncountable.
â Leonard Blackburn
Aug 8 at 17:49
2
Well, I'm just trying to help clarify the question. I haven't given a solution. I think the hints and discussion below do that for you. But the point is you cannot assume anything about B except: 1. its elements are all positive real numbers and 2. if you take a finite number of elements from B and add them up then you get a sum less than two. Given any such B you have to prove that B is either finite or countable.
â Leonard Blackburn
Aug 8 at 17:59
1
Here is a proof that B cannot be the open interval (0,1). If it were then these numbers would be in B: 0.5, 0.51, 0.52, and 0.53. These numbers add up to more than 2. Do you see? You could do something similar to show that B cannot be (0,a) for any positive a. But you have to show more than that. You have to show B is finite or countable.
â Leonard Blackburn
Aug 8 at 18:00
 |Â
show 6 more comments
4
It sounds like you are confused about what the question is asking. It is not saying that B is the set of positive real numbers; it is a set of positive real numbers. This means that B is a subset of the set of all positive real numbers.
â Leonard Blackburn
Aug 8 at 17:23
2
No. B is some unspecified subset of the positive real numbers such that if you add up a finite number of the elements of B the sum is always less than 2. For example, B could be 0.1, 0.2, 0.5, 1.1 or B could be the infinite set 1, 1/2, 1/4, 1/8, 1/16, .... Note that B cannot be 1, 1.4, 1.8 and B cannot be 1, 1/2, 1/3, 1/4, ...
â Leonard Blackburn
Aug 8 at 17:47
2
B does not have to be an interval (0,a). In fact, it cannot be such an interval. Your job is to prove that B must be finite or countable. Intervals are uncountable.
â Leonard Blackburn
Aug 8 at 17:49
2
Well, I'm just trying to help clarify the question. I haven't given a solution. I think the hints and discussion below do that for you. But the point is you cannot assume anything about B except: 1. its elements are all positive real numbers and 2. if you take a finite number of elements from B and add them up then you get a sum less than two. Given any such B you have to prove that B is either finite or countable.
â Leonard Blackburn
Aug 8 at 17:59
1
Here is a proof that B cannot be the open interval (0,1). If it were then these numbers would be in B: 0.5, 0.51, 0.52, and 0.53. These numbers add up to more than 2. Do you see? You could do something similar to show that B cannot be (0,a) for any positive a. But you have to show more than that. You have to show B is finite or countable.
â Leonard Blackburn
Aug 8 at 18:00
4
4
It sounds like you are confused about what the question is asking. It is not saying that B is the set of positive real numbers; it is a set of positive real numbers. This means that B is a subset of the set of all positive real numbers.
â Leonard Blackburn
Aug 8 at 17:23
It sounds like you are confused about what the question is asking. It is not saying that B is the set of positive real numbers; it is a set of positive real numbers. This means that B is a subset of the set of all positive real numbers.
â Leonard Blackburn
Aug 8 at 17:23
2
2
No. B is some unspecified subset of the positive real numbers such that if you add up a finite number of the elements of B the sum is always less than 2. For example, B could be 0.1, 0.2, 0.5, 1.1 or B could be the infinite set 1, 1/2, 1/4, 1/8, 1/16, .... Note that B cannot be 1, 1.4, 1.8 and B cannot be 1, 1/2, 1/3, 1/4, ...
â Leonard Blackburn
Aug 8 at 17:47
No. B is some unspecified subset of the positive real numbers such that if you add up a finite number of the elements of B the sum is always less than 2. For example, B could be 0.1, 0.2, 0.5, 1.1 or B could be the infinite set 1, 1/2, 1/4, 1/8, 1/16, .... Note that B cannot be 1, 1.4, 1.8 and B cannot be 1, 1/2, 1/3, 1/4, ...
â Leonard Blackburn
Aug 8 at 17:47
2
2
B does not have to be an interval (0,a). In fact, it cannot be such an interval. Your job is to prove that B must be finite or countable. Intervals are uncountable.
â Leonard Blackburn
Aug 8 at 17:49
B does not have to be an interval (0,a). In fact, it cannot be such an interval. Your job is to prove that B must be finite or countable. Intervals are uncountable.
â Leonard Blackburn
Aug 8 at 17:49
2
2
Well, I'm just trying to help clarify the question. I haven't given a solution. I think the hints and discussion below do that for you. But the point is you cannot assume anything about B except: 1. its elements are all positive real numbers and 2. if you take a finite number of elements from B and add them up then you get a sum less than two. Given any such B you have to prove that B is either finite or countable.
â Leonard Blackburn
Aug 8 at 17:59
Well, I'm just trying to help clarify the question. I haven't given a solution. I think the hints and discussion below do that for you. But the point is you cannot assume anything about B except: 1. its elements are all positive real numbers and 2. if you take a finite number of elements from B and add them up then you get a sum less than two. Given any such B you have to prove that B is either finite or countable.
â Leonard Blackburn
Aug 8 at 17:59
1
1
Here is a proof that B cannot be the open interval (0,1). If it were then these numbers would be in B: 0.5, 0.51, 0.52, and 0.53. These numbers add up to more than 2. Do you see? You could do something similar to show that B cannot be (0,a) for any positive a. But you have to show more than that. You have to show B is finite or countable.
â Leonard Blackburn
Aug 8 at 18:00
Here is a proof that B cannot be the open interval (0,1). If it were then these numbers would be in B: 0.5, 0.51, 0.52, and 0.53. These numbers add up to more than 2. Do you see? You could do something similar to show that B cannot be (0,a) for any positive a. But you have to show more than that. You have to show B is finite or countable.
â Leonard Blackburn
Aug 8 at 18:00
 |Â
show 6 more comments
3 Answers
3
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oldest
votes
up vote
3
down vote
accepted
Not only can we show that it's countable, it's not too difficult to construct an enumeration: given an element $b$, just count how many other elements of $B$ are larger. This has to be a finite integer, since if there were an infinite number of elements greater than $b$, then the sum of $n$ such elements would be greater than $nb$, so picking an $n>2/b$ would give a sum greater than 2.
add a comment |Â
up vote
17
down vote
Hint 1: How many elements of $B$ can be in the set $[2,infty)$?
Hint 2: How many elements of $B$ can be in the set $[1,2)$?
Hint 3: How many elements of $B$ can be in the set $[0.5,1)$?
24
At this rate you are going to need countably many hints :)
â Arnaud Mortier
Aug 8 at 17:04
9
@ArnaudMortier Can't he give a hint schema?
â saulspatz
Aug 8 at 17:07
1) One $x = 2$, 2) set $x_1=1$ so one for sure, then I don't know how many $x_2,x_3,...x_n$ to choose so that they sum up to $1$. 3) The same, I don't know.
â Sargis Iskandaryan
Aug 8 at 17:10
2
Obviously $Bcap[1,infty)$ has at most two elements, because if $x,y,zge1$ then $x+y+z>2$. Similarly $Bcap[2/n,infty)$ has at most $n$ elements. (So $B$ is the union of countably many finite sets...)
â David C. Ullrich
Aug 8 at 17:17
1
No, we certainly can't write $B$ as $(0,a)$. If $B=(0,a)$ then there are finite subsets of $B$ with sum larger than $2$.
â David C. Ullrich
Aug 8 at 18:56
 |Â
show 8 more comments
up vote
1
down vote
For any $ain B$, define $B_aequivbin B:a<b$. The cardinality of $B_a$ must be less than $lceil2/arceil$, otherwise the sum of any $lceil2/arceil$ elements of $B_a$ would exceed 2. That is, $sum_i=1^lceil2/arceilx_i>sum_i=1^lceil2/arceila=alceil2/arceilge2$ for any sequence $x_i$ in $B_a$, contradicting the requirements on $B$.
Define the function $f:BrightarrowmathbbN$ as $f(a)=left|B_aright|$.
Homework: If you can show that $f$ is one-to-one then you can conclude that $|B|le|mathbbN|$. That is, $B$ is either finite or countable.
Credit goes to Acccumulation for the outline of this proof.
add a comment |Â
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
Not only can we show that it's countable, it's not too difficult to construct an enumeration: given an element $b$, just count how many other elements of $B$ are larger. This has to be a finite integer, since if there were an infinite number of elements greater than $b$, then the sum of $n$ such elements would be greater than $nb$, so picking an $n>2/b$ would give a sum greater than 2.
add a comment |Â
up vote
3
down vote
accepted
Not only can we show that it's countable, it's not too difficult to construct an enumeration: given an element $b$, just count how many other elements of $B$ are larger. This has to be a finite integer, since if there were an infinite number of elements greater than $b$, then the sum of $n$ such elements would be greater than $nb$, so picking an $n>2/b$ would give a sum greater than 2.
add a comment |Â
up vote
3
down vote
accepted
up vote
3
down vote
accepted
Not only can we show that it's countable, it's not too difficult to construct an enumeration: given an element $b$, just count how many other elements of $B$ are larger. This has to be a finite integer, since if there were an infinite number of elements greater than $b$, then the sum of $n$ such elements would be greater than $nb$, so picking an $n>2/b$ would give a sum greater than 2.
Not only can we show that it's countable, it's not too difficult to construct an enumeration: given an element $b$, just count how many other elements of $B$ are larger. This has to be a finite integer, since if there were an infinite number of elements greater than $b$, then the sum of $n$ such elements would be greater than $nb$, so picking an $n>2/b$ would give a sum greater than 2.
answered Aug 8 at 18:25
Acccumulation
5,2542515
5,2542515
add a comment |Â
add a comment |Â
up vote
17
down vote
Hint 1: How many elements of $B$ can be in the set $[2,infty)$?
Hint 2: How many elements of $B$ can be in the set $[1,2)$?
Hint 3: How many elements of $B$ can be in the set $[0.5,1)$?
24
At this rate you are going to need countably many hints :)
â Arnaud Mortier
Aug 8 at 17:04
9
@ArnaudMortier Can't he give a hint schema?
â saulspatz
Aug 8 at 17:07
1) One $x = 2$, 2) set $x_1=1$ so one for sure, then I don't know how many $x_2,x_3,...x_n$ to choose so that they sum up to $1$. 3) The same, I don't know.
â Sargis Iskandaryan
Aug 8 at 17:10
2
Obviously $Bcap[1,infty)$ has at most two elements, because if $x,y,zge1$ then $x+y+z>2$. Similarly $Bcap[2/n,infty)$ has at most $n$ elements. (So $B$ is the union of countably many finite sets...)
â David C. Ullrich
Aug 8 at 17:17
1
No, we certainly can't write $B$ as $(0,a)$. If $B=(0,a)$ then there are finite subsets of $B$ with sum larger than $2$.
â David C. Ullrich
Aug 8 at 18:56
 |Â
show 8 more comments
up vote
17
down vote
Hint 1: How many elements of $B$ can be in the set $[2,infty)$?
Hint 2: How many elements of $B$ can be in the set $[1,2)$?
Hint 3: How many elements of $B$ can be in the set $[0.5,1)$?
24
At this rate you are going to need countably many hints :)
â Arnaud Mortier
Aug 8 at 17:04
9
@ArnaudMortier Can't he give a hint schema?
â saulspatz
Aug 8 at 17:07
1) One $x = 2$, 2) set $x_1=1$ so one for sure, then I don't know how many $x_2,x_3,...x_n$ to choose so that they sum up to $1$. 3) The same, I don't know.
â Sargis Iskandaryan
Aug 8 at 17:10
2
Obviously $Bcap[1,infty)$ has at most two elements, because if $x,y,zge1$ then $x+y+z>2$. Similarly $Bcap[2/n,infty)$ has at most $n$ elements. (So $B$ is the union of countably many finite sets...)
â David C. Ullrich
Aug 8 at 17:17
1
No, we certainly can't write $B$ as $(0,a)$. If $B=(0,a)$ then there are finite subsets of $B$ with sum larger than $2$.
â David C. Ullrich
Aug 8 at 18:56
 |Â
show 8 more comments
up vote
17
down vote
up vote
17
down vote
Hint 1: How many elements of $B$ can be in the set $[2,infty)$?
Hint 2: How many elements of $B$ can be in the set $[1,2)$?
Hint 3: How many elements of $B$ can be in the set $[0.5,1)$?
Hint 1: How many elements of $B$ can be in the set $[2,infty)$?
Hint 2: How many elements of $B$ can be in the set $[1,2)$?
Hint 3: How many elements of $B$ can be in the set $[0.5,1)$?
answered Aug 8 at 17:04
vadim123
74.2k895185
74.2k895185
24
At this rate you are going to need countably many hints :)
â Arnaud Mortier
Aug 8 at 17:04
9
@ArnaudMortier Can't he give a hint schema?
â saulspatz
Aug 8 at 17:07
1) One $x = 2$, 2) set $x_1=1$ so one for sure, then I don't know how many $x_2,x_3,...x_n$ to choose so that they sum up to $1$. 3) The same, I don't know.
â Sargis Iskandaryan
Aug 8 at 17:10
2
Obviously $Bcap[1,infty)$ has at most two elements, because if $x,y,zge1$ then $x+y+z>2$. Similarly $Bcap[2/n,infty)$ has at most $n$ elements. (So $B$ is the union of countably many finite sets...)
â David C. Ullrich
Aug 8 at 17:17
1
No, we certainly can't write $B$ as $(0,a)$. If $B=(0,a)$ then there are finite subsets of $B$ with sum larger than $2$.
â David C. Ullrich
Aug 8 at 18:56
 |Â
show 8 more comments
24
At this rate you are going to need countably many hints :)
â Arnaud Mortier
Aug 8 at 17:04
9
@ArnaudMortier Can't he give a hint schema?
â saulspatz
Aug 8 at 17:07
1) One $x = 2$, 2) set $x_1=1$ so one for sure, then I don't know how many $x_2,x_3,...x_n$ to choose so that they sum up to $1$. 3) The same, I don't know.
â Sargis Iskandaryan
Aug 8 at 17:10
2
Obviously $Bcap[1,infty)$ has at most two elements, because if $x,y,zge1$ then $x+y+z>2$. Similarly $Bcap[2/n,infty)$ has at most $n$ elements. (So $B$ is the union of countably many finite sets...)
â David C. Ullrich
Aug 8 at 17:17
1
No, we certainly can't write $B$ as $(0,a)$. If $B=(0,a)$ then there are finite subsets of $B$ with sum larger than $2$.
â David C. Ullrich
Aug 8 at 18:56
24
24
At this rate you are going to need countably many hints :)
â Arnaud Mortier
Aug 8 at 17:04
At this rate you are going to need countably many hints :)
â Arnaud Mortier
Aug 8 at 17:04
9
9
@ArnaudMortier Can't he give a hint schema?
â saulspatz
Aug 8 at 17:07
@ArnaudMortier Can't he give a hint schema?
â saulspatz
Aug 8 at 17:07
1) One $x = 2$, 2) set $x_1=1$ so one for sure, then I don't know how many $x_2,x_3,...x_n$ to choose so that they sum up to $1$. 3) The same, I don't know.
â Sargis Iskandaryan
Aug 8 at 17:10
1) One $x = 2$, 2) set $x_1=1$ so one for sure, then I don't know how many $x_2,x_3,...x_n$ to choose so that they sum up to $1$. 3) The same, I don't know.
â Sargis Iskandaryan
Aug 8 at 17:10
2
2
Obviously $Bcap[1,infty)$ has at most two elements, because if $x,y,zge1$ then $x+y+z>2$. Similarly $Bcap[2/n,infty)$ has at most $n$ elements. (So $B$ is the union of countably many finite sets...)
â David C. Ullrich
Aug 8 at 17:17
Obviously $Bcap[1,infty)$ has at most two elements, because if $x,y,zge1$ then $x+y+z>2$. Similarly $Bcap[2/n,infty)$ has at most $n$ elements. (So $B$ is the union of countably many finite sets...)
â David C. Ullrich
Aug 8 at 17:17
1
1
No, we certainly can't write $B$ as $(0,a)$. If $B=(0,a)$ then there are finite subsets of $B$ with sum larger than $2$.
â David C. Ullrich
Aug 8 at 18:56
No, we certainly can't write $B$ as $(0,a)$. If $B=(0,a)$ then there are finite subsets of $B$ with sum larger than $2$.
â David C. Ullrich
Aug 8 at 18:56
 |Â
show 8 more comments
up vote
1
down vote
For any $ain B$, define $B_aequivbin B:a<b$. The cardinality of $B_a$ must be less than $lceil2/arceil$, otherwise the sum of any $lceil2/arceil$ elements of $B_a$ would exceed 2. That is, $sum_i=1^lceil2/arceilx_i>sum_i=1^lceil2/arceila=alceil2/arceilge2$ for any sequence $x_i$ in $B_a$, contradicting the requirements on $B$.
Define the function $f:BrightarrowmathbbN$ as $f(a)=left|B_aright|$.
Homework: If you can show that $f$ is one-to-one then you can conclude that $|B|le|mathbbN|$. That is, $B$ is either finite or countable.
Credit goes to Acccumulation for the outline of this proof.
add a comment |Â
up vote
1
down vote
For any $ain B$, define $B_aequivbin B:a<b$. The cardinality of $B_a$ must be less than $lceil2/arceil$, otherwise the sum of any $lceil2/arceil$ elements of $B_a$ would exceed 2. That is, $sum_i=1^lceil2/arceilx_i>sum_i=1^lceil2/arceila=alceil2/arceilge2$ for any sequence $x_i$ in $B_a$, contradicting the requirements on $B$.
Define the function $f:BrightarrowmathbbN$ as $f(a)=left|B_aright|$.
Homework: If you can show that $f$ is one-to-one then you can conclude that $|B|le|mathbbN|$. That is, $B$ is either finite or countable.
Credit goes to Acccumulation for the outline of this proof.
add a comment |Â
up vote
1
down vote
up vote
1
down vote
For any $ain B$, define $B_aequivbin B:a<b$. The cardinality of $B_a$ must be less than $lceil2/arceil$, otherwise the sum of any $lceil2/arceil$ elements of $B_a$ would exceed 2. That is, $sum_i=1^lceil2/arceilx_i>sum_i=1^lceil2/arceila=alceil2/arceilge2$ for any sequence $x_i$ in $B_a$, contradicting the requirements on $B$.
Define the function $f:BrightarrowmathbbN$ as $f(a)=left|B_aright|$.
Homework: If you can show that $f$ is one-to-one then you can conclude that $|B|le|mathbbN|$. That is, $B$ is either finite or countable.
Credit goes to Acccumulation for the outline of this proof.
For any $ain B$, define $B_aequivbin B:a<b$. The cardinality of $B_a$ must be less than $lceil2/arceil$, otherwise the sum of any $lceil2/arceil$ elements of $B_a$ would exceed 2. That is, $sum_i=1^lceil2/arceilx_i>sum_i=1^lceil2/arceila=alceil2/arceilge2$ for any sequence $x_i$ in $B_a$, contradicting the requirements on $B$.
Define the function $f:BrightarrowmathbbN$ as $f(a)=left|B_aright|$.
Homework: If you can show that $f$ is one-to-one then you can conclude that $|B|le|mathbbN|$. That is, $B$ is either finite or countable.
Credit goes to Acccumulation for the outline of this proof.
edited Aug 9 at 11:06
answered Aug 9 at 3:13
Matt
482412
482412
add a comment |Â
add a comment |Â
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4
It sounds like you are confused about what the question is asking. It is not saying that B is the set of positive real numbers; it is a set of positive real numbers. This means that B is a subset of the set of all positive real numbers.
â Leonard Blackburn
Aug 8 at 17:23
2
No. B is some unspecified subset of the positive real numbers such that if you add up a finite number of the elements of B the sum is always less than 2. For example, B could be 0.1, 0.2, 0.5, 1.1 or B could be the infinite set 1, 1/2, 1/4, 1/8, 1/16, .... Note that B cannot be 1, 1.4, 1.8 and B cannot be 1, 1/2, 1/3, 1/4, ...
â Leonard Blackburn
Aug 8 at 17:47
2
B does not have to be an interval (0,a). In fact, it cannot be such an interval. Your job is to prove that B must be finite or countable. Intervals are uncountable.
â Leonard Blackburn
Aug 8 at 17:49
2
Well, I'm just trying to help clarify the question. I haven't given a solution. I think the hints and discussion below do that for you. But the point is you cannot assume anything about B except: 1. its elements are all positive real numbers and 2. if you take a finite number of elements from B and add them up then you get a sum less than two. Given any such B you have to prove that B is either finite or countable.
â Leonard Blackburn
Aug 8 at 17:59
1
Here is a proof that B cannot be the open interval (0,1). If it were then these numbers would be in B: 0.5, 0.51, 0.52, and 0.53. These numbers add up to more than 2. Do you see? You could do something similar to show that B cannot be (0,a) for any positive a. But you have to show more than that. You have to show B is finite or countable.
â Leonard Blackburn
Aug 8 at 18:00