>>16771472
It kind of reminds me of this. I otherwise love this book but this is one of the most egregiously wrong proofs I've seen: the idea is to show that the composition of a path and its path inverse is homotopic to identity.

I think what kind of happens is an author 'knows' a result to be true and goes through the motions of the immediate idea that comes to mind when they verify it. They are less critical about whether the proof is accurate since well the result is so obviously true. I kind of feel that is what is going on here with do Carmo.

This to me actually showcases the importance of rigor, ie even if you explain the basic idea and handwave it, may be totally wrong since you've missed some details.