Single variable calculus question

[Rule]

This was given to me on a timed calculus test with 7 questions (several years ago). 

Suppose satisfies for all . 

Show that



Notation: The notation may be confusing, so I will explain.  f is continuous in the closed interval from 0 to 1, and the absolute value of its derivative is less than A in this interval.  We want to show the inequality for all natural numbers n (n = 0, 1, 2, ...).  It is unlikely you will find a solution to this problem on the internet.  Do not get hung up on small technicalities, you do not need to be very careful about definitions to develop a general idea of how to solve this.  I will post two different proofs after some attempts have been made. 

Edit: I will fix the images later.  For now, just right click and "view image".