This was given to me on a timed calculus test with 7 questions (several years ago).
Suppose (http://www.valhallalegends.com/rule/calcquestions/calc2/func.gif) satisfies (http://www.valhallalegends.com/rule/calcquestions/calc2/cond.gif) for all (http://www.valhallalegends.com/rule/calcquestions/calc2/tin.gif).
Show that
(http://www.valhallalegends.com/rule/calcquestions/calc2/maineq.gif)
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".