Direct Proof

is a method of proving. We use it to prove statements of the form: if p then q. The method is like this:

Take an original statement p which we assume to be true, and use it to show directly with other facts as necessary that another statement q is true.

Example
To prove: If n is an odd integer then n2 is also an odd integer.

Proof:

By definition, an odd integer is defined as: 2k + 1 for some integer k. Thus: n2 = (2k + 1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1

As we can see, the last expression is also of the form: 2k + 1 for some integer k. Thus n2 is also an odd integer when n is an odd integer.