Correctness of proof by induction

Question

Suppose a person states the following: $n^2 = (n * n), \forall n > 0$. One can check such equality by saying, via proof by induction, that:

• for $n := 0:\ 0^2 = (0 * 0)$;
• for $n := 1:\ 1^2 = (1 * 1)$;

• and for $n := n + 1:$

  $\ (n + 1)^2 = ((n + 1) * (n + 1))$

  $n^2 + 2n + 1 = n^2 + 2n + 1$

As far as I understand, this states that:

• $P(n)$ is valid for some $n$ within a range, say integers greater than 0;
• $P(n)$ must be valid for the successor of any $n$ within the previous range;

Of course, it is not commonly necessary to prove such things as $n^2 = (n * n)$. And, proving it seems rather like considering that, eventually, there could be a mysterious value for $n$ that would make $n^2 \neq (n * n)$.

First question: is this interpretation correct?

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Now, if one could prove $P(n): n^2 = (n * n)$ via the base and the induction cases:

• $P(n)$ is valid for $n := 0$;
• $P(n)$ is valid for $n := n + 1$;

One should also be allowed to prove it by saying that:

• $P(n)$ is valid for $n := 0$;
• $P(n)$ is valid for $n := n + 2$;
• and via $P(n)$ is valid for $n := n + 3$;
• and via $P(n)$ is valid for $n := n + 4$;
• and, in fact, via $P(n)$ is valid for $n := n + m,\ \forall m > 0$.

Second question: is this inference correct?

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So, if we are fine thus far, it should be the case that, eventually, a mysterious value $m$ could make proof by induction fail. And here comes the final question.

Third Question: how to prove the correctness of proof by induction?