Organizing Proofs.
A proof should consist of a sequence of statements (preferably each statement being on a separate line):
Typically, each proof starts with an assumption, and finishes with a conclusion.
Each statement should be a consequence of the previous statements in the proof, together with the axioms or properties of the system you are working in. (For example, if you are proving something about the real numbers, you get to use the axioms for ordered fields).
It is important that at each line, you say how you are deducing the statement from the previous lines &/or the axioms.
Example:
Using the axioms for an ordered field as in the handout, prove that for all a, b, c, in an ordered field, (a+b)c=ac+bc.
Proof: (a+b)c = c(a+b) by axiom 1 (commutativity)
= ca+cb by axiom 3 (distributivity)
= ac + bc by axiom 1 (commutativity)
Proofs by Induction.
To prove a statement (predicate) P(n) about natural numbers n, by induction, you need to:
P(n+1) Þ P(n)
Some no-nos.