Routine Exercise 4 (Working with the Definitions) Let f(n) = n³ . For each of the functions g given below, formally prove, directly from the definition, that g (f). Each of your 6 separate proofs must clearly state N and c as in the definitions of O() and (), and must clearly explain the algebra involved in any inequalities.

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)