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?