1. **Problem Statement:** Prove that there are infinitely many even numbers. 2. **Definition:** An even number is any integer that can be written as $2k$ where $k$ is an integer. 3. **Proof by Contradiction:** Assume there are only finitely many even numbers. Let the largest even number be $2n$ for some integer $n$. 4. **Construct a Larger Even Number:** Consider $2(n+1)$. Since $n+1$ is an integer, $2(n+1)$ is an even number. 5. **Contradiction:** $2(n+1)$ is greater than $2n$, contradicting the assumption that $2n$ is the largest even number. 6. **Conclusion:** Therefore, there cannot be a largest even number, and hence, there are infinitely many even numbers.