Proof by cases is a way of proving something by looking at different situations, or cases, and showing that they all lead to the same conclusion. Think about it like this: you want to prove that all animals have legs. To do this, you might consider two cases: animals with legs, and animals without legs. For animals with legs, you can use examples like dogs, cats, and horses to show that they all have legs. For animals without legs, you can use examples like fish and snakes to show that they are not actually classified as animals. Once you've shown that both cases lead to the same conclusion (that animals have legs), you've proved your point. Proof by cases is a useful method because it allows you to take into account different possibilities and still come to a definitive answer.