# Inequality Induction problem 1

Here's the claim again, for reference:

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

Induction Hypothesis: Suppose that $$2^n n! < (2n)!$$ is true for $$n = 2, 3, \ldots, k$$ where k is an integer $$\geq 2$$.
In particular, the induction hypothesis tells us that $$2^{k}k! < (2k)!$$.
We need to show that $$2^{k+1} (k+1)! < (2(k+1))!$$