What's wrong with this (fake) proof that $ n=1\forall n\in \Bbb N$?
Base case: $n=1$ true.
$n-2,n-1<n+1\implies n-1=n-2\implies n+1=n=1$.
From the principle of induction it follows that $n=1 \forall n\in \Bbb N._\square$
This "proof" comes from a book of Enzo Gentile.
So, the base case $n=1$ is true. Then, to show it for $2$, we start out with the statement $$-1,0<2$$ and then concludes that $-1=0$, despite that neither $-1$ nor $0$ are covered in this induction. That is, it reaches "before" the base cases. If we had $-1=1$ and $0=1$, this would be valid. But those are, of course, false.