The proof would be correct but for sloppy statements which are correct in a "You know what I mean way" but as stated are totally false. And it is important that we see them so we don't fall for them and make invalid proofs. (Although this particular proof is valid.)
1) "Suppose now that P(n) is true for every n∈N."
That is assuming what we want to prove. We can't do that. And if we did assume such a thing we couldn't accept any results from it.
What should be sai is "Suppose that P(n) is true for some non-specified n". At this point it could be true for $n = 32$ but not $n=17$. It is true for $n=1$ so we do know it is true for at least one so we can state that. But we don't want to assume $n$ is $1$. Just that $n$ is some unspecified value so that $P(n)$ is true.
2)"Thus, since P(1) is true, and P(n) and P(n+1) are true, the statement P(n) is true for all natural numbers n, by the principle of induction."
Well, not not quite. Suppose may statement was. "P(n) = N is not a square of a prime" and I note P(1) is true (1 is not prime). And P(n=6) is true and P(n+1 = 7) is true and conclude therefore that this is true for all natural numbers by induction.
What should have been stated was "Since P(1) is true, and P(n) being true will imply P(n+1) is true, then P(n) is true for all $n$ by induction."
It is essential that P(n+1) being true is true via a direct inference for a non-specific $n$ where $P(n)$ is true. If $P(n+1)$ is true via a pure coincidence or from the specifics of $n$, we can't make any general conclusions.
(Also, nitpicky, the statement "P(1), P(n) and P(n+1) are true" is a bit meaningless. P(n) is true? For what $n$, when did we ever show that? At one point we assumed it but we never showed it. And how can we say $P(n)$ is true if we never stated what $n$ is? What is meant is $P(n)$ is true for some $n$. But then $P(n+1)$ is true for some $n+1$ and those two statements mean nothing unless there is some implication that for any true P(n) then P(n+1) is true.)
With those corrections the proof is completely valid.