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For all natural numbers n ≥ 4, 2n ≤ n!. Proof. We have 24 = 16...

For all natural numbers n ≥ 4, 2n ≤ n!.
Proof. We have 24 = 16 and 4! = 24, so the statement is true for n = 4. Assume that 2n < n! for some n. Then

2n+1 =2(2n)<2(n!)≤(n+1)(n!)=(n+1)!
so 2n+1 < (n + 1)!. Thus, by the PMI, the statement is true for all n ≥ 4.

This proof is incorrect. How do you prove it the correct way? The formula of the proof was said to be incorrect but I still don't understand.

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