Exhaustive Proof
An exhaustive proof is a type of proof that covers all possible cases. It is a method of proving a statement by considering and addressing each and every possible scenario or case.

Explanation
When dealing with a situation where there are multiple possibilities or cases, an exhaustive proof ensures that each and every case is considered and proven. This type of proof is particularly useful when the number of cases is finite and manageable.

In an exhaustive proof, the following steps are typically followed:
1. Identify all possible cases: The first step is to identify and list down all the possible cases or scenarios that need to be considered in order to prove the statement.
2. Address each case individually: For each case, the proof is carried out separately, providing evidence and reasoning specific to that particular case.
3. Cover all cases: It is important to ensure that all cases are covered and addressed in the proof. This means that no case should be left out or overlooked.
4. Present the proof: The proof is presented in a clear and organized manner, demonstrating the validity of the statement for each case.

Advantages of Exhaustive Proof
- Completeness: An exhaustive proof ensures that all possible cases are considered, providing a comprehensive and complete analysis of the statement.
- Confidence: By addressing each case individually, an exhaustive proof instills confidence in the validity of the statement, as it covers all possible scenarios.
- Accuracy: The thorough examination of all cases in an exhaustive proof minimizes the chances of errors or oversights.

Example
Let's consider a statement: "All prime numbers are odd." To prove this statement using an exhaustive proof, we would consider all possible cases:
1. Case 1: Prime number is 2 (the only even prime number). In this case, the statement is false, as 2 is not odd.
2. Case 2: Prime number is any other odd number. In this case, the statement is true, as all odd prime numbers are indeed odd.

By considering and addressing both cases, we have covered all possibilities and proven the statement. This is an example of an exhaustive proof.

Conclusion
Exhaustive proof is a method of proof that covers all possible cases. It is particularly useful when dealing with a finite number of cases and ensures the completeness and accuracy of the proof. By considering and addressing each case individually, an exhaustive proof provides confidence in the validity of the statement.