Perhaps you want to know how mathematicians prove their theorems. There are some books devoted to just this subject. The best book which I am aware of is the book of Polya, "How to solve it?". You may find it in the library. Polya states (and I agree with him) that every process of proving is a sequence of questions and answers. There are some elementary questions and one ``only" has to arrange these questions in a proper order. If you want to be a mathematician you have to know what these questions are, of course. Here I'd like to show you how I solved one problem, using the method of Polya.
Proof.
Question 1. What things this statement is about?
Answer. This statement is about numbers a,b,c
Question 2. What do we know about these numbers?
Answer. We know that a divides b and b divides c.
Question 3. What does it mean?
Answer.This means:
a) there exists an integer x such that b=ax.
b) there exists an integer y such that c=by.
(You must always denote all things which appear in your proof).
Question 4. What do we have to prove?
Answer.We have to prove that a divides c.
Question 5. What does it mean?
Answer. This means that there exists an integer z such that c=za.
Question 6. Can we rewrite the statement which we are proving using this new information?
Answer. Yes, we can. We just have to find z such that c=za given the facts that b=ax and c=by.
Question 7. Can we find this z?
Answer. Yes, we can: z=xy. Indeed, az=axy=(ax)y=by=c.
The proof is complete.