The idea of proof by induction is as follows.
Suppose that you want to prove that a statement $P(n)$ is true for all integers $n\ge 0$.
You start by proving $P(0)$ is true
assume that $P(k)$ is true for some $k\ge 0$
prove that it must then follow that $P(k+1)$ is true, in other words you prove $P(k)$ implies $P(k+1)$.
The idea is that, as you know $P(0)$ is true, since you checked it, you can use the $P(k)$ to $P(k+1)$ step to prove that $P(1)$ is true, then use the $P(k)$ to $P(k+1)$ step again to prove $P(2)$ is true and so on, and thereby you prove $P(n)$ is true for all $n$.
Which one of the following is not a mistake?
You should assume, as above, that you are trying to prove a formula or statement $P(n)$ for ALL integers $n\ge 0$.