How do you prove a mathematical statement via contradiction?

For a proof via contradiction, you would start by assuming the statement is actually false¹ (even if the given statement seems inherently correct, it is essential for the proof). Now that you have a basis for your argument, you can make use of mathematical steps and logic² to show that the assumption leads to an impossible situation/conclusion³ (either a contradiction of the assumption or a contradiction of a fact that is known to be true). You are now able to conclude that your assumption was incorrect, and thus the original statement is true⁴.For example, to prove 'If x² is even, then x is also even' via contradiction: 1) Assume the statement is actually false. Suppose that x² is an even integer, and that x is in fact an odd integer. 2) Use mathematical steps and logic. We can represent x, being odd, as x=2n+1 where n is any integer. Now x²=(2n+1)²=4n²+4n+1. By factorising this result we end up with x²= 2k+1, where k=2n²+2n. 3) A contradiction or an impossible situation. Having shown x²=2k+1, it is impossible for x² to be an even integer. This contradicts our original statement. 4) Conclusion. Our assumption (If x² is even, then x is odd) has been shown to be impossible, thus proving by contradiction that the original statement (If x² is even, then x is also even) must be correct/true.