Here's the claim again, for reference:
Claim: For all positive integers \(n \geq 2\), \(2^n n! < (2n)!\)
Base: You should be checking the claim at n=2.
Induction Hypothesis: Suppose that \(2^n n! < (2n)!\) is true for \(n = 2, 3, \ldots, k\) where k is an integer \(\geq 2\).
Rest of induction step:
In particular, the induction hypothesis tells us that \(2^{k}k! < (2k)!\).
We need to show that \(2^{k+1} (k+1)! < (2(k+1))!\)