Major remark. The problem says "for all integers n >= 0", so the base case should be for n = 0.

(Pretty) minor remarks. In the beginning of the induction step, it is good to write the restriction on k, i.e., k >= 0. I guess, here the fact that 2 < k + 3 is obvious, but when I am trying to evaluate it for the first time, I immediately want to see what k is.

Here you start with what you need to prove and end with something similar to the induction hypothesis. This is a natural record of simplifying the inequality one has to prove. However, when proofs are written in their final form, they usually go from assumptions through intermediate statements to the final result.2^(k+1) < (k+1+2)!

(2^k)2 < (k+3)!

2(2^k) < (k+3)(k+2)!

2^k< [(k+3)/2](k+2)!