The inductive step is being obscured (intentionally of course in the article to illustrate how the error can hide) behind the n.
The inductive step here is not true for all n, n + 1 pairs.
It is only true for all n, n + 1 pairs given that n is greater than or equal to 2.
The error is to drop that >= condition. Hence the inductive step is wrong as stated in the article (for all n), and its proof is erroneous.
Again while changing the base case can "fix" the argument in that it no longer proves an falsehood, without altering the statement of the inductive step it is still not a valid argument and only happens to prove a true statement "by chance."
If the base case was the error in the sense of proving a falsehood, then it would be possible to have an unsound inductive argument by choosing the wrong base case, but that's impossible. Rather the wrong base case (but correct inductive step) simply means you end up unable to complete your proof.
The inductive step here is not true for all n, n + 1 pairs.
It is only true for all n, n + 1 pairs given that n is greater than or equal to 2.
The error is to drop that >= condition. Hence the inductive step is wrong as stated in the article (for all n), and its proof is erroneous.
Again while changing the base case can "fix" the argument in that it no longer proves an falsehood, without altering the statement of the inductive step it is still not a valid argument and only happens to prove a true statement "by chance."
If the base case was the error in the sense of proving a falsehood, then it would be possible to have an unsound inductive argument by choosing the wrong base case, but that's impossible. Rather the wrong base case (but correct inductive step) simply means you end up unable to complete your proof.