A proof by contraposition

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Question from Rahul:

I am not able to understand the following,
To prove that if for all e>0, |x|<e, then x=0, the contrapositive approach is used as follows,
assume x =/= (not equal to) 0 and hence we can take e=|x|>0, then |x|>=e. I understand the approach very well but I do not understand why if |x|=e then |x|>=e. If it is so then why not |x|=<e etc.
Thanks in advance!'
Rahul.

Hi Rahul,

To repeat your argument you say

Suppose $x \neq 0$ and take $e = |x| > 0.$ Then $|x| \not< e,$ and this completes the contrapositive. You have chosen to express $|x| \not< e$ as $|x| \ge e.$ The two are equivalent so it doesn't matter which way you say it. Since $|x| = e$ you can also say $|x| \leq e$ which is $|x| \not>e,$ but that doesn't complete the proof.

Harley

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