Please make handwriting Legible. Thanks in advance.

This project is intended to give you some simple, concrete experience with the formal definitions of O and O. Let f(n) = n. For each of the functions g given below, formally prove, directly from the definition, that e O(g). Each of your 6 separate proofs must clearly state N and c as in the definition of O() and must clearly show the algebra proving your inequalities. Make sure that you have some reason for choosing N and c, rather than just guessing and bluffing that the corresponding inequality is true (I'll be more skeptical of your inequalities if you haven't picked N and c sensibly). a. g(n) = 1000n3 b. g(n) = n3 + 1000n2 c. g(n) = n3 1000n2 (assume n > 1000 for this to be a legitimate function for our purposes)