In mathematics we make assertions about a system whether it be a number
system or something more abstract such as a group or linear space.
An assertion not known to be true or false is called a hypothesis or
conjecture. Prior to 1995, a famous conjecture was Fermat's Last
Theorem. It stated that for an integer 
there are no positive integer
solutions to the equation 
.
The process of establishing the truth of an assertion is called a proof.
Once a conjecture has been shown to be a true statement we label it as a
lemma, theorem or corollary. We think of a lemma as a
result which is used primarily to prove a more important result (i.e. a
theorem), and a corollary as a special case or consequence of a theorem.
For example in calculus, we could think of Maclaurin's Theorem as a corollary
to Taylor's Theorem.
In these notes we are concerned with techniques that may be used to prove a result and provide a tonic to the student's malady on proofs namely ``I don't know where to start''. It is probably impossible to teach how to prove something and the best one can offer is a catalog of types of proof along with examples. By reading proofs, the student can often gain insight as to how to prove their own particular result. Once they have gained some experience, they might then be ready for more complicated proofs. What is certain is that there is no cook book solution to obtaining a proof. We recommend that you read the notes on logic before proceeding.