Is the following statement true?
Two real numbers a and b are equal iff for every ε > 0, |a − b| < ε.
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.
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2otherwise $|a-b| < \frac{|a-b|}{2}$ is false – mathworker21 Aug 31 '19 at 14:02
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2The only nonnegative real number smaller than every positive numebr is $0$. – saulspatz Aug 31 '19 at 14:04
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@mathworker21: That's not how it works. ε can be any constant, e.g. 0.1 or 3.14, but it is not a function f(a,b). – MSalters Aug 31 '19 at 22:40
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@MSalters that is how it works. you're fixing $a$ and $b$ and asking if it holds that $|a-b| < \epsilon$ for all $\epsilon > 0$. Included in "for all" is $\epsilon = \frac{|a-b|}{2}$. – mathworker21 Aug 31 '19 at 22:44
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@mathworker21: Fair point, after fixing a and b any function f(a,b) will also have a fixed value. – MSalters Aug 31 '19 at 22:47
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Does this answer your question? Intuition: If $a\leq b+\epsilon$ for all $\epsilon>0$ then $a\leq b$? – user1110341 Jun 10 '23 at 06:54
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Quick beginner guide for asking a well-received question + please avoid "no clue" questions. – Anne Bauval Jun 14 '23 at 11:42
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@user1110341 Again: that post, which you proposed as a duplicate of many questions, cannot be considered as such. It was about intuition, not proof. Moreover, though related, the question is not identical. – Anne Bauval Jun 14 '23 at 11:43
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The statement is correct. Here is a proof of the converse.
Suppose that $a\neq b$.
Then $\epsilon:=|a-b|>0$ but we do not have $|a-b|<\epsilon$.
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This question goes against our quality standard policy. Instead of posting an answer, you should encourage the person who posted the question to improve it. – Anne Bauval Jun 14 '23 at 11:44
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Yes, it is true because the condition is satisfied for every $\epsilon>0,$ no matter how little. When we say a number is less than such an $\epsilon,$ we simply mean that the number vanishes, or is $0.$
Allawonder
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This question goes against our quality standard policy. Instead of posting an answer, you should encourage the person who posted the question to improve it. – Anne Bauval Jun 14 '23 at 11:44
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This is outright proven in Understanding Analysis (2016 2 edn). pp 9 - 10.
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This question goes against our quality standard policy. Instead of posting an answer, you should encourage the person who posted the question to improve it. – Anne Bauval Jun 14 '23 at 11:44