Proof by Cases

Proof by Cases is the concept of splitting a statement into parts to make it simpler to prove. Its analogous to a divide and conquer strategy. Each case can be proven with varying methods and sometimes could differ between the cases.

Example:

Prove that for all integers, n² + 5n is even.

Proof:

Odd and even integers behave differently with respect to what the result is. So, we have to separate this into two cases: even integers and odd integers. Since, every integer is either even or odd this will satisfy our proposition.

Case 1:

Suppose n is even. Theorem 1 tells us that n² is even. With 5 being odd and n being even Theorem 1 tells us 5*n is even. Thus, n² + 5*n is even, by Theorem 1.

Case 2:

Suppose n is odd. By Theorem 1, n² is odd. With n and 5 being odd, Theorem 1 tells us 5*n is also odd. Therefore, n² + 5*n is even, by Theorem 1.

Thus we have shown that if n is odd or even n² + 5*n is even. So, our proposition is true.