There are many techniques that can be used to prove the statements. It is often not obvious at the beginning which one to use, although with a bit of practice, we may be able to give an "educated guess" and hopefully reach the required conclusion. It is important to notice that there is no one ideal proof - a theorem can be established using different techniques and none of them will be better or worse (as long as they are all valid).
We can divide the techniques into two groups; direct proofs and indirect proofs. Direct proof assumes a given hypothesis, or any other known statement, and then logically deduces a conclusion.
Indirect proof, also called proof by contradiction, assumes the hypothesis (if given) together with a negation of a conclusion to reach the contradictory statement. It is often equivalent to proof by contrapositive, though it is subtly different (see the examples). Both direct and indirect proofs may also include additional tools to reach the required conclusions, namely proof by cases or mathematical induction.
