Hello everyone! I started on this mathematical induction problem and I am not sure how to finish it past a certain point. Here it is:
Question: Prove that if S is any finite set with |S| = n, then |S x S x S x S x S| ≤ |P(S)|,for all n ≥ N where N is some constant, the minimum value of which you must discover and use as the basis for your proof.
Because the Cartesian product grows at the rate of 2n, we must show that n5≤ 2n for n ≥ N, and find what N is.
Base Case: n = 23.
235 ≤ 223= 6436343 ≤ 8388608. Base case proven. (This is not true for any n less than 23 since 225 = 5153632 and 222 = 4194304, and 5153632 is not less than 4194304.)
Inductive Hypothesis: Assume for some k ≥ 23 that k5 ≤ 2k.
Inductive Step: (k+1)5≤ 2k+1
(k+1)5= k5 + 5k4 + 10k3 + 10k2 + 5k + 1
I am not sure where to go from here, to be honest. Would anyone care to help? It would be much appreciated!