1. **State the problem:** Prove by mathematical induction that for all even natural numbers $n$, the statement $P(n)$ holds. Since the problem statement is incomplete, let's assume a common induction problem for even numbers, for example: "Prove that for all even natural numbers $n$, $n^2$ is even." 2. **Base case:** Let $n=2$, the smallest even natural number. Calculate $2^2 = 4$, which is even. So, $P(2)$ is true. 3. **Inductive hypothesis:** Assume that for some even natural number $k$, $k^2$ is even. That is, assume $P(k)$ is true. 4. **Inductive step:** We need to prove $P(k+2)$ is true, i.e., $(k+2)^2$ is even. Calculate: $$(k+2)^2 = k^2 + 4k + 4$$ By the inductive hypothesis, $k^2$ is even. Since $k$ is even, $4k$ is also even. And $4$ is even. Sum of even numbers is even, so $(k+2)^2$ is even. 5. **Conclusion:** By the principle of mathematical induction, $n^2$ is even for all even natural numbers $n$. **Final answer:** The statement is proven true by induction for all even natural numbers $n$.