Mixing
Analysis. Supremum and infimum.
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Let $A,B subsetmathbb{R}$ and $C ={x+y | x ∈ A,y ∈ B}$. How are the numbers $inf A$, $inf B$, and $inf C$ related? How are the numbers $sup A$, $sup B$, and $sup C$ related?
real-analysis analysis supremum-and-infimum
edited Feb 3 at 6:03
Don Fanucci
1,325521
asked Feb 3 at 5:56
John LongJohn Long
11
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add a comment |
$begingroup$
Let $A,B subsetmathbb{R}$ and $C ={x+y | x ∈ A,y ∈ B}$. How are the numbers $inf A$, $inf B$, and $inf C$ related? How are the numbers $sup A$, $sup B$, and $sup C$ related?
real-analysis analysis supremum-and-infimum
edited Feb 3 at 6:03
Don Fanucci
1,325521
asked Feb 3 at 5:56
John LongJohn Long
11
$endgroup$
$begingroup$
Welcome to math.stackexchange. You will find your experience much improved if instead of simply quoting the problem you are struggling with (or have been assigned to do), you provide both context (where did the problem come from? What is your level?) and also explain what you have done to try to solve it and where you are running into difficulties. That way, the answers can be given at the appropriate level and not cover material you already know (or fail to solve your difficulties). Otherwise, you may find the question getting closed for lack of effort on your part.
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– Arturo Magidin
Feb 3 at 6:11
$begingroup$
Do you have any guesses? Maybe try some examples first, and see if you can come up with a conjecture to prove.
$endgroup$
– Theo Bendit
Feb 3 at 6:17
add a comment |
$begingroup$
Let $A,B subsetmathbb{R}$ and $C ={x+y | x ∈ A,y ∈ B}$. How are the numbers $inf A$, $inf B$, and $inf C$ related? How are the numbers $sup A$, $sup B$, and $sup C$ related?
real-analysis analysis supremum-and-infimum
edited Feb 3 at 6:03
Don Fanucci
1,325521
asked Feb 3 at 5:56
John LongJohn Long
11
$endgroup$
Let $A,B subsetmathbb{R}$ and $C ={x+y | x ∈ A,y ∈ B}$. How are the numbers $inf A$, $inf B$, and $inf C$ related? How are the numbers $sup A$, $sup B$, and $sup C$ related?
real-analysis analysis supremum-and-infimum
real-analysis analysis supremum-and-infimum
edited Feb 3 at 6:03
Don Fanucci
1,325521
asked Feb 3 at 5:56
John LongJohn Long
11
edited Feb 3 at 6:03
Don Fanucci
1,325521
asked Feb 3 at 5:56
John LongJohn Long
11
edited Feb 3 at 6:03
Don Fanucci
1,325521
edited Feb 3 at 6:03
Don Fanucci
1,325521
edited Feb 3 at 6:03
Don Fanucci
1,325521
1,325521
asked Feb 3 at 5:56
John LongJohn Long
11
asked Feb 3 at 5:56
John LongJohn Long
11
asked Feb 3 at 5:56
John LongJohn Long
11
11
$begingroup$
Welcome to math.stackexchange. You will find your experience much improved if instead of simply quoting the problem you are struggling with (or have been assigned to do), you provide both context (where did the problem come from? What is your level?) and also explain what you have done to try to solve it and where you are running into difficulties. That way, the answers can be given at the appropriate level and not cover material you already know (or fail to solve your difficulties). Otherwise, you may find the question getting closed for lack of effort on your part.
$endgroup$
– Arturo Magidin
Feb 3 at 6:11
$begingroup$
Do you have any guesses? Maybe try some examples first, and see if you can come up with a conjecture to prove.
$endgroup$
– Theo Bendit
Feb 3 at 6:17
add a comment |
$begingroup$
Welcome to math.stackexchange. You will find your experience much improved if instead of simply quoting the problem you are struggling with (or have been assigned to do), you provide both context (where did the problem come from? What is your level?) and also explain what you have done to try to solve it and where you are running into difficulties. That way, the answers can be given at the appropriate level and not cover material you already know (or fail to solve your difficulties). Otherwise, you may find the question getting closed for lack of effort on your part.
$endgroup$
– Arturo Magidin
Feb 3 at 6:11
$begingroup$
Do you have any guesses? Maybe try some examples first, and see if you can come up with a conjecture to prove.
$endgroup$
– Theo Bendit
Feb 3 at 6:17
$begingroup$
Welcome to math.stackexchange. You will find your experience much improved if instead of simply quoting the problem you are struggling with (or have been assigned to do), you provide both context (where did the problem come from? What is your level?) and also explain what you have done to try to solve it and where you are running into difficulties. That way, the answers can be given at the appropriate level and not cover material you already know (or fail to solve your difficulties). Otherwise, you may find the question getting closed for lack of effort on your part.
$endgroup$
– Arturo Magidin
Feb 3 at 6:11
$begingroup$
Welcome to math.stackexchange. You will find your experience much improved if instead of simply quoting the problem you are struggling with (or have been assigned to do), you provide both context (where did the problem come from? What is your level?) and also explain what you have done to try to solve it and where you are running into difficulties. That way, the answers can be given at the appropriate level and not cover material you already know (or fail to solve your difficulties). Otherwise, you may find the question getting closed for lack of effort on your part.
$endgroup$
– Arturo Magidin
Feb 3 at 6:11
$begingroup$
Do you have any guesses? Maybe try some examples first, and see if you can come up with a conjecture to prove.
$endgroup$
– Theo Bendit
Feb 3 at 6:17
$begingroup$
Do you have any guesses? Maybe try some examples first, and see if you can come up with a conjecture to prove.
$endgroup$
– Theo Bendit
Feb 3 at 6:17
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
From the properties of the infimum,
$$ inf(C) = inf(A) + inf (B) $$
To prove, you can use the theorem that if a number $z$ is a lower bound of $X$ and there is a sequence of numbers of the set $X$ approaching $z$, then $z$ is the infimum of $X$.
Same applies for the supremum.
edited Feb 3 at 6:56
Chase Ryan Taylor
4,46221531
answered Feb 3 at 6:49
Hello Darkness my old friendHello Darkness my old friend
525
$endgroup$
1
$begingroup$
Nice answer! Welcome to MSE $ddotsmile$
$endgroup$
– Chase Ryan Taylor
Feb 3 at 6:54
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
From the properties of the infimum,
$$ inf(C) = inf(A) + inf (B) $$
To prove, you can use the theorem that if a number $z$ is a lower bound of $X$ and there is a sequence of numbers of the set $X$ approaching $z$, then $z$ is the infimum of $X$.
Same applies for the supremum.
edited Feb 3 at 6:56
Chase Ryan Taylor
4,46221531
answered Feb 3 at 6:49
Hello Darkness my old friendHello Darkness my old friend
525
$endgroup$
1
$begingroup$
Nice answer! Welcome to MSE $ddotsmile$
$endgroup$
– Chase Ryan Taylor
Feb 3 at 6:54
add a comment |
$begingroup$
From the properties of the infimum,
$$ inf(C) = inf(A) + inf (B) $$
To prove, you can use the theorem that if a number $z$ is a lower bound of $X$ and there is a sequence of numbers of the set $X$ approaching $z$, then $z$ is the infimum of $X$.
Same applies for the supremum.
edited Feb 3 at 6:56
Chase Ryan Taylor
4,46221531
answered Feb 3 at 6:49
Hello Darkness my old friendHello Darkness my old friend
525
$endgroup$
1
$begingroup$
Nice answer! Welcome to MSE $ddotsmile$
$endgroup$
– Chase Ryan Taylor
Feb 3 at 6:54
add a comment |
$begingroup$
From the properties of the infimum,
$$ inf(C) = inf(A) + inf (B) $$
To prove, you can use the theorem that if a number $z$ is a lower bound of $X$ and there is a sequence of numbers of the set $X$ approaching $z$, then $z$ is the infimum of $X$.
Same applies for the supremum.
edited Feb 3 at 6:56
Chase Ryan Taylor
4,46221531
answered Feb 3 at 6:49
Hello Darkness my old friendHello Darkness my old friend
525
$endgroup$
From the properties of the infimum,
$$ inf(C) = inf(A) + inf (B) $$
To prove, you can use the theorem that if a number $z$ is a lower bound of $X$ and there is a sequence of numbers of the set $X$ approaching $z$, then $z$ is the infimum of $X$.
Same applies for the supremum.
edited Feb 3 at 6:56
Chase Ryan Taylor
4,46221531
answered Feb 3 at 6:49
Hello Darkness my old friendHello Darkness my old friend
525
edited Feb 3 at 6:56
Chase Ryan Taylor
4,46221531
edited Feb 3 at 6:56
Chase Ryan Taylor
4,46221531
edited Feb 3 at 6:56
Chase Ryan Taylor
4,46221531
4,46221531
answered Feb 3 at 6:49
Hello Darkness my old friendHello Darkness my old friend
525
answered Feb 3 at 6:49
Hello Darkness my old friendHello Darkness my old friend
525
answered Feb 3 at 6:49
Hello Darkness my old friendHello Darkness my old friend
525
525
1
$begingroup$
Nice answer! Welcome to MSE $ddotsmile$
$endgroup$
– Chase Ryan Taylor
Feb 3 at 6:54
add a comment |
1
$begingroup$
Nice answer! Welcome to MSE $ddotsmile$
$endgroup$
– Chase Ryan Taylor
Feb 3 at 6:54
1
1
$begingroup$
Nice answer! Welcome to MSE $ddotsmile$
$endgroup$
– Chase Ryan Taylor
Feb 3 at 6:54
$begingroup$
Nice answer! Welcome to MSE $ddotsmile$
$endgroup$
– Chase Ryan Taylor
Feb 3 at 6:54
add a comment |
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$begingroup$
Welcome to math.stackexchange. You will find your experience much improved if instead of simply quoting the problem you are struggling with (or have been assigned to do), you provide both context (where did the problem come from? What is your level?) and also explain what you have done to try to solve it and where you are running into difficulties. That way, the answers can be given at the appropriate level and not cover material you already know (or fail to solve your difficulties). Otherwise, you may find the question getting closed for lack of effort on your part.
$endgroup$
– Arturo Magidin
Feb 3 at 6:11
$begingroup$
Do you have any guesses? Maybe try some examples first, and see if you can come up with a conjecture to prove.
$endgroup$
– Theo Bendit
Feb 3 at 6:17