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

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=4

1) If n = 4 ⇒2 (4) < 4! ⇒2 (4) < 24 ⇒8 < 24.

Therefore the statement holds for n=4

2) Now we assume that the statement is valid for n = k

⇒2k < k!

3) Now we will prove the statement holds for n = k + 1

We 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 that

2 (k + 1) < (k+1) (k) (k-1) ... (3) (2) (1)

2 < (k) (k-1) ... (3) (2) (1)

which is clearly true since k ≥4

Therefore the statement n ≥4, 2n < n! is true.