proof of growth of exponential function
In this proof, we first restrict to when and are integers and only later lift this restricton.
Let be an integer, let be real, and let be an integer.
Consider the following inequality
If , then we have
Define to be the greater of and ; when , we have
Rewrite as follows when :
By the inequality established above, each term in the product will be bounded by , hence
Since , it is also the case that , hence we have the inequality
Combining the last two inequalities yields the following:
From this, it follows that when and are integers.
Now we lift the restriction that be an integer. Since the power function is increasing, , so we have for real values of as well.
To lift the restriction on , let us write where is an integer and . Then we have
If , then . Since . Hence, for all real , we have
From this inequality, it follows that for real values of as well.
|Title||proof of growth of exponential function|
|Date of creation||2013-03-22 15:48:36|
|Last modified on||2013-03-22 15:48:36|
|Last modified by||rspuzio (6075)|