You can multiply both sides of an equation or an inequality by the same number, but your last equality is still wrong. However, it leads to the right idea.
We know (by assumption) that . How does one obtain from and (n + 2)! from (n + 1)! ? is obtained by multiplying by 2, so let's multiply both sides of by 2. Then how does 2(k + 1)! compare with (k + 2)! ?