proof of Cauchy condition for limit of function
The forward direction is . Assume that . Then given there is a such that
Now for and we have
We prove the reverse by contradiction. Assume that the condition holds. Now suppose that does not exist. This means that for any and any sufficiently small then for any there is such that . For any such choose such that and put then substituting in the condition with we get . A contradiction.
|Title||proof of Cauchy condition for limit of function|
|Date of creation||2013-03-22 18:59:08|
|Last modified on||2013-03-22 18:59:08|
|Last modified by||puff (4175)|