I still don’t quite understand. The inductive step shows n + 1 → n right? However with any positive base case b, n → n + 1 isn’t certain for any integers above b, only below it.

Say you’ve proved the case for n = 3 and that n + 1 → n. Then you’ve proven that 2 + 1 → 3, and by induction 1 + 1 → 2, However you’ve never proven it for n = 4 because n → n + 1 has never been established.

Or am I missing something here?

EDIT: I’ve seen in other posts that this the problem with OP is that it hides the transitivity of the operation. In fact the failure of the proof was that it proved transitivity with a false premise. If transitivity was true, then using n + 1 → n is just fine. The Wikipedia article for this statement is actually a lot clearer. https://en.wikipedia.org/wiki/All_horses_are_the_same_color