continuity of natural power
Proof. Let be any positive number. Denote and . Then identically
But since and also , so each summand in the parentheses is at most equal to , and since there are summands, the sum is at most equal to . Thus we get
We may choose ; this implies
The right hand side of this inequality is less than as soon as we still require
This means that the power function is continuous at the point .
Note. Another way to prove the theorem is to use induction on and the rule 2 in limit rules of functions.
|Title||continuity of natural power|
|Date of creation||2013-03-22 15:39:25|
|Last modified on||2013-03-22 15:39:25|
|Last modified by||pahio (2872)|