discrete mathematics - How can mathematical induction prove something?

I am learning mathematical induction, and the concept still does not fit in my mind. I just cannot understand how I can prove something just by:

1) basis: calculating whether it fits for the minimal $n$, where $n$ belongs to $N$.

2) inductive step: I JUST assume that the statement that I set at the start works for any $n\leq m$, then it also works for $(m+1)$

On what basis can I say that the statement is proven? Based on the fact that I have found the formula from the first part of the inductive step in the formula of the second part of the inductive step, substituting the first one for the second and getting the same result as I assumed in the inductive step?

I cannot get it. Maybe I am misunderstanding the whole concept of mathematical induction. If that is the truth, then I am sorry.

Can anyone explain to me in human language why I can say that a statement is proven when I perform mathematical induction on that statement?

Answer

I'll explain weak induction, which is probably what you're learning. Let's say you want to prove a statement for $n\geq N$. Say this statement is $P(n)$.

1) Basis step: Prove the statement for $n=N$, i.e. $P(N)$ is true.

2) Inductive step: Suppose $P(m)$ is true for some $m\geq N$. Then you use this assumption to prove that $P(m+1)$ is true.

How the two steps work is as follows. In the inductive step you should have proved $P(m+1)$ is true using $P(m)$ is true, without explicitly stating what $m$ is (in other words, you DON'T substitute it with a number, say 100 or something).

This means that no matter what $m$ is, it will always be the case that when $P(m)$ is true, then $P(m+1)$ is true, and this holds for all $m\geq N$.

Now in the basis step, you have proved $P(N)$ is true. By the preceding paragraph, this means $P(N+1)$ is true. Then $P(N+2)$ is true. Then $P(N+3)$ is true, and so on. So $P(n)$ is true for all $n\geq N$.