Answer:
There are several ways to write a proof, depending on the level of rigor and formality required. Some common methods include:
1- Direct proof: A direct proof is a logical argument that begins with the premises and uses logical deduction to arrive at the conclusion.
2- Proof by contradiction: A proof by contradiction starts by assuming the opposite of what needs to be proven and then showing that this leads to a contradiction.
3- Proof by contrapositive: A proof by contrapositive is a proof that starts by assuming the negation of the desired conclusion and then showing that this leads to the negation of the premises.
4- Proof by induction: A proof by induction is a proof that starts by showing that a statement is true for the first few natural numbers, and then showing that if it is true for a particular natural number, it must also be true for the next natural number.
5- Proof by construction: A proof by construction is a proof that shows the existence of a mathematical object by giving an explicit construction of it.
6- Proof by exhaustion: A proof by exhaustion is a proof that considers all possible cases and shows that in each case the statement is true.
7- Proof by casework: A proof by casework is a proof that is split into several cases, each of which is considered separately.
8- Proof by diagram: A proof by diagram is a proof that uses a visual representation to show that a statement is true.
These are some of the common ways of writing proof, and there could be other ways as well. It is important to use the most appropriate method for the problem at hand, and to clearly explain and justify each step of the proof.