Is the following statement true: two real numbers a and b are equal iff for every ε > 0, |a − b| < ε.Are the following theorems' converses also true?How do I show convergence here?which of the following statement are true?Let $a_n$ be a sequence of real numbers. which one of the following is always true?Is it always true that $f$ is continuous at $c$ given the following conditionsQuick Answer Verification: When are two Norms/Operator Norms EqualThen which of the following statement(s) is(are) true? about uniform convergenceLet $A$ be the set of all functions $f:mathbbRtomathbbR$ that satisfy the following two properties.Which of the following statements are true for $A=tsin(frac1t) $?
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Is the following statement true: two real numbers a and b are equal iff for every ε > 0, |a − b|
Are the following theorems' converses also true?How do I show convergence here?which of the following statement are true?Let $a_n$ be a sequence of real numbers. which one of the following is always true?Is it always true that $f$ is continuous at $c$ given the following conditionsQuick Answer Verification: When are two Norms/Operator Norms EqualThen which of the following statement(s) is(are) true? about uniform convergenceLet $A$ be the set of all functions $f:mathbbRtomathbbR$ that satisfy the following two properties.Which of the following statements are true for $A=tsin(frac1t) $?
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I got that if a and b are equal then |a-b|=0 which is less than ε.
But I'm not sure if the converse also holds.
real-analysis
asked 8 hours ago
s1234s1234
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add a comment |
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I got that if a and b are equal then |a-b|=0 which is less than ε.
But I'm not sure if the converse also holds.
real-analysis
asked 8 hours ago
s1234s1234
111 bronze badge
New contributor
s1234 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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otherwise $|a-b| < frac2$ is false
$endgroup$
– mathworker21
8 hours ago
$begingroup$
The only nonnegative real number smaller than every positive numebr is $0$.
$endgroup$
– saulspatz
8 hours ago
add a comment |
$begingroup$
I got that if a and b are equal then |a-b|=0 which is less than ε.
But I'm not sure if the converse also holds.
real-analysis
asked 8 hours ago
s1234s1234
111 bronze badge
New contributor
s1234 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
I got that if a and b are equal then |a-b|=0 which is less than ε.
But I'm not sure if the converse also holds.
real-analysis
real-analysis
asked 8 hours ago
s1234s1234
111 bronze badge
New contributor
s1234 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 8 hours ago
s1234s1234
111 bronze badge
New contributor
s1234 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 8 hours ago
s1234s1234
111 bronze badge
New contributor
s1234 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 8 hours ago
s1234s1234
111 bronze badge
asked 8 hours ago
s1234s1234
111 bronze badge
111 bronze badge
New contributor
s1234 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
s1234 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$begingroup$
otherwise $|a-b| < frac2$ is false
$endgroup$
– mathworker21
8 hours ago
$begingroup$
The only nonnegative real number smaller than every positive numebr is $0$.
$endgroup$
– saulspatz
8 hours ago
add a comment |
$begingroup$
otherwise $|a-b| < frac2$ is false
$endgroup$
– mathworker21
8 hours ago
$begingroup$
The only nonnegative real number smaller than every positive numebr is $0$.
$endgroup$
– saulspatz
8 hours ago
$begingroup$
otherwise $|a-b| < frac2$ is false
$endgroup$
– mathworker21
8 hours ago
$begingroup$
otherwise $|a-b| < frac2$ is false
$endgroup$
– mathworker21
8 hours ago
$begingroup$
The only nonnegative real number smaller than every positive numebr is $0$.
$endgroup$
– saulspatz
8 hours ago
$begingroup$
The only nonnegative real number smaller than every positive numebr is $0$.
$endgroup$
– saulspatz
8 hours ago
add a comment |
1 Answer
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$begingroup$
You are correct. Here is a proof of the converse.
Suppose that $aneq b$.
Then $epsilon:=|a-b|>0$ but we do not have $|a-b|<epsilon$.
answered 8 hours ago
drhabdrhab
112k5 gold badges49 silver badges141 bronze badges
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1 Answer
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$begingroup$
You are correct. Here is a proof of the converse.
Suppose that $aneq b$.
Then $epsilon:=|a-b|>0$ but we do not have $|a-b|<epsilon$.
answered 8 hours ago
drhabdrhab
112k5 gold badges49 silver badges141 bronze badges
$endgroup$
add a comment |
$begingroup$
You are correct. Here is a proof of the converse.
Suppose that $aneq b$.
Then $epsilon:=|a-b|>0$ but we do not have $|a-b|<epsilon$.
answered 8 hours ago
drhabdrhab
112k5 gold badges49 silver badges141 bronze badges
$endgroup$
add a comment |
$begingroup$
You are correct. Here is a proof of the converse.
Suppose that $aneq b$.
Then $epsilon:=|a-b|>0$ but we do not have $|a-b|<epsilon$.
answered 8 hours ago
drhabdrhab
112k5 gold badges49 silver badges141 bronze badges
$endgroup$
You are correct. Here is a proof of the converse.
Suppose that $aneq b$.
Then $epsilon:=|a-b|>0$ but we do not have $|a-b|<epsilon$.
answered 8 hours ago
drhabdrhab
112k5 gold badges49 silver badges141 bronze badges
edited 7 hours ago
answered 8 hours ago
drhabdrhab
112k5 gold badges49 silver badges141 bronze badges
answered 8 hours ago
drhabdrhab
112k5 gold badges49 silver badges141 bronze badges
answered 8 hours ago
drhabdrhab
112k5 gold badges49 silver badges141 bronze badges
112k5 gold badges49 silver badges141 bronze badges
add a comment |
add a comment |
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$begingroup$
otherwise $|a-b| < frac2$ is false
$endgroup$
– mathworker21
8 hours ago
$begingroup$
The only nonnegative real number smaller than every positive numebr is $0$.
$endgroup$
– saulspatz
8 hours ago