The second step in proof by induction is to:
Prove that if the statement is true for some integer $n=k$, where $k\ge n_0$ then it is also true for the next larger integer, $n=k+1$
My question is about the "if"-statement. Can we just assume that indeed the statement is true? If we assume it, then the proof works... but isn't that similar to the following "proof":
Let $N$ be the largest positive integer.
Since $1$ is a positive integer, we must have $N\ge1$.
Since $N^2$ is a positive integer, it cannot exceed the largest positive integer.
Therefore, $N^2\le N$ and so $N^2-N\le0$.
Thus, $N(N-1)\le0$ and we must have $N-1\le0$.
Therefore, $N\le1$. Since also $N\ge1$, we have $N=1$.
Therefore, $1$ is the largest positive integer.
The only thing that is wrong with this "proof" is that we falsely assume there actually exists a largest positive integer.
So both in the above case and in proof by induction we do an assumption. In the second case the assumption leads to a false conclusion. What is the difference with proof by induction? Why is doing the assumption that the hypothesis is actually true valid here and why doesn't it lead to a similar contradiction?
EDIT: the "proof" above is not mine, it is taken from Calculus a Complete Course 8th edition as an example of why existence proofs are important.