1. The problem is to understand how to prove a mathematical statement or theorem. 2. A proof is a logical argument that demonstrates the truth of a statement using definitions, axioms, and previously established results. 3. The general approach to proof involves several steps: - State the theorem or statement clearly. - List known facts, definitions, and assumptions. - Use logical reasoning and mathematical operations to connect the assumptions to the conclusion. 4. Important rules include: - Each step must follow logically from the previous ones. - Use valid mathematical operations and properties. - Avoid circular reasoning. 5. For example, to prove that the sum of two even numbers is even: - Let the two even numbers be $2a$ and $2b$, where $a$ and $b$ are integers. - Their sum is $2a + 2b = 2(a + b)$. - Since $a + b$ is an integer, $2(a + b)$ is even by definition. 6. This shows the sum of two even numbers is even, completing the proof. 7. Proofs can be direct, by contradiction, by induction, or by contrapositive depending on the problem. 8. Practice and familiarity with logical reasoning improve proof skills.