Give the Proofs in Mathematics ?

1 Two-column deductive proof

Proof:
Statements                                                              Reasons
* Start with given conditions or                                  * Given information
with a conclusion based on a diagram                           * Definitions
and label all the statements sequentially.                       * Postulates
* Develop a chain of reasoning. Write                            * Previously proved theorems
each logical step as a separate statement                      * Algebraic properties
with a corresponding reason.
* Continue the above steps until a statement
corresponding to the conclusion of the
original argument is reached.

The two-column deductive proof uses a direct method to prove an argument in a finite number of sequential steps. In geometry, the two-column deductive proof is the most common and powerful method and is usually referred to as a "formal" proof.

2 Indirect method of proof

1. From the given information, list all the possibilities including the given conclusion.

2. Prove that all possibilities other than the given conclusion are false.

3. Therefore, only the given conclusion is true.

3 Counterexample proof