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# This step is not the source of the fallacy.

This step is correctly stating what it means to prove a statement of the form "for every X, statement S is true": you consider an arbitrary X, and show that S is true for it.

It is stated in step 4 that we are trying to prove that, in every group of k+1 people, everyone has the same age.

In general, the way to prove that every "whatchamacallit" has a certain property is to consider a typical whatchamacallit x, and show that x has the desired property. As long as the argument works no matter which whatchamacallit is being considered, we will have proven that all whatchamacallits have the property.

In our case, a "whatchamacallit" means a group of k+1 people. So, we consider a typical watchamacallit, G, and we will show that G has the desired property, namely that everyone in G is the same age.

As long as our argument works no matter which group G of k+1 people is being considered, we will have proven that, in every group of k+1 people, everyone has the same age.

Why don't you go back to the list of steps in the proof and see if you can identify which one is wrong, now that you know it isn't this one?