Prove that for all n ≥ 4 the inequality 2n < n! holds.

Answer:For all n ≥ 4, 2n < n! Step-by-step explanation:Let's use the induction method to prove this statement. In the induction method, first we prove the statement for n=41) If n = 4 ⇒2(4) < 4! ⇒2(4) < 24 ⇒8 < 24.Therefore the statement holds for n=42) Now we assume that the statement is valid for  n = k ⇒2k < k!3) Now we will prove the statement holds for n = k +1We will prove that 2(k + 1) < (k +1)!(k + 1)! = (k+1) (k) (k-1) .... (3) (2) (1)If the statement is valid for k + 1, then it would mean that2 (k + 1) < (k+1) (k) (k-1) ... (3) (2) (1)2 < (k) (k-1).... (3) (2) (1)which is clearly true since k ≥4Therefore the statement n ≥4, 2n < n! is true.