1. **Stating the problem:** We want to understand what a proof is and see an example of finding a proof in mathematics. 2. **What is a proof?** A proof is a logical argument that shows a statement is always true using definitions, axioms, and previously proven theorems. 3. **Example problem:** Prove that the sum of two even numbers is always even. 4. **Formula and definitions:** An even number can be written as $2k$ where $k$ is an integer. 5. **Proof steps:** - Let the two even numbers be $2a$ and $2b$, where $a$ and $b$ are integers. - Their sum is $2a + 2b$. - Factor the sum: $$2a + 2b = 2(a + b)$$ - Since $a + b$ is an integer, call it $c = a + b$. - Therefore, the sum is $2c$, which is even by definition. 6. **Conclusion:** We have shown that the sum of two even numbers is always even by expressing them in terms of integers and factoring. This is a simple example of a proof in algebra.