I'm learning induction. Spot the error in this induction:
Let $P(n): 1 = 2 = 3 = \dots = n$
It's true for $n = 1$, because $1 = 1$, then let's assume $P(n)$ is true *, so we must show that $P(n+1)$ is true. Summing $1$ on every element we will have: $2 = 3 = 4 \dots = n = n + 1$ **
Then, with * and **, we have that $P(n+1)$ is true.
I'm not enterly sure where's the error, but I think the error is in **, because we don't have $1 = n+1$, or maybe, because at the end we have $n = n+1$ that is false.
Well, if there's any missconcept or anything wrong in my thoughts please let me know.
This is a disguised version of the famous "all horses are the same colour" proof. The induction step does not actually go through in every case; in particular, it fails if $n = 2$. As written, you have $1 = 1$, adding $1$ to this equation gives you... $2 = 2$. Not $2 = 1$ like you need to complete the argument.