Step 2 states that the statement is true for n=1.
Step 3 states that the only other thing needed is to prove that, whenever the statement is true for one number (say n = k), it must also be true for the next number (that is, for n = k+1).
Steps 4 through 5 state that the only thing needed to accomplish this goal is to show that everyone in an arbitrary group G of k+1 people has the same age, under the assumption that, in every group of k people, everyone has the same age.
Step 13 states that this has been accomplished.
Therefore, from these steps, it follows that the statement is true for all n.
Of course, we know it's not, so that means the fallacy must lie in one of the earlier steps.