Inequality Induction problem 1

Here's the claim again, for reference:

Claim: For all positive integers \(n \geq 2\), \(2^n n! < (2n)!\)

Check your outline

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 \(2^{k+1} (k+1)! < (2(k+1))!\)