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