Big Oh and Omega notation complexity proof

Question

1. Prove that n³ is not in O(n²)
2. Prove that n³ is not in OMEGA(n⁴)

Answer 1

1. Suppose that n³ is in O(n²), then there exists some pair of positive constants c and n₀ such that n³ ≤ cn² for all n ≥ n₀, but for any constant c this is trivially false when n > c, thus we have a contradiction.

2. Suppose that n³ is in Ω(n⁴), then there exists some pair of positive constants c and n₀ such that n³ ≥ cn⁴ for all n ≥ n₀, but for any constant c, this is trivially false when n > max(1,1/c), thus we have a contradiction.