Write a general outline for a proof

- A theorem is a statement that can be proven true or false.…A proof consists of a series of steps,…each of which follows logically from assumptions…or from previously proven statements…whose final step should result in a statement…of the theorem being proven.…Axioms are statements assumed to be true,…or previously proven theorems.…An example is the definition of even and odd integers.…An even integer is any integer…that can be represented by two times x,…where x is also an integer.…

10 is two times five,…200 is two times 100, etcetera.…An odd number is represented by any number…such that it can be written as two times x plus one.…So 11 is two times five, plus one.…Proofs fall into two categories, direct and indirect proofs.…With an indirect proof, instead of proving…that the conclusion is true, you start by assuming…that the opposite is true.…

Then, using deductive reasoning…to lead to a contradiction, you can prove…the original hypotheses is true.…Proof by contradiction is another type of indirect proof,…and requires a counter-example.…